On the spectrum of the double-layer operator on locally-dilation-invariant Lipschitz domains

Abstract

We say that $\Gamma$, the boundary of a bounded Lipschitz domain, is locally dilation invariant if, at each $x\in \Gamma$, $\Gamma$ is either locally $C^1$ or locally coincides (in some coordinate system centred at x) with a Lipschitz graph $\Gamma _x$ such that $\Gamma _x=\alpha _x\Gamma _x$, for some $\alpha _x\in (0,1)$. In this paper we study, for such $\Gamma$, the essential spectrum of $D_\Gamma$, the double-layer (or Neumann–Poincaré) operator of potential theory, on $L^2(\Gamma )$. We show, via localisation and Floquet–Bloch-type arguments, that this essential spectrum is the union of the spectra of related continuous families of operators $K_t$, for $t\in [-\pi ,\pi ]$; moreover, each $K_t$ is compact if $\Gamma$ is $C^1$ except at finitely many points. For the 2D case where, additionally, $\Gamma$ is piecewise analytic, we construct convergent sequences of approximations to the essential spectrum of $D_\Gamma$; each approximation is the union of the eigenvalues of finitely many finite matrices arising from Nyström-method approximations to the operators $K_t$. Through error estimates with explicit constants, we also construct functionals that determine whether any particular locally-dilation-invariant piecewise-analytic $\Gamma$ satisfies the well-known spectral radius conjecture, that the essential spectral radius of $D_\Gamma$ on $L^2(\Gamma )$ is $<1/2$ for all Lipschitz $\Gamma$. We illustrate this theory with examples; for each we show that the essential spectral radius is $<1/2$, providing additional support for the conjecture. We also, via new results on the invariance of the essential spectral radius under locally-conformal $C^{1,\beta }$ diffeomorphisms, show that the spectral radius conjecture holds for all Lipschitz curvilinear polyhedra.

1 Introduction

Given a bounded Lipschitz domain¹$\Omega _- \subset \mathbb {R}^d, d \ge 2$, with boundary $\Gamma$ and outward-pointing unit normal vector n, the interior and exterior Dirichlet and Neumann problems for Laplace’s equation (posed in $\Omega _-$ and in $\Omega _+:=\mathbb {R}^d{\setminus } \overline{\Omega _-}$, respectively), can be reformulated as boundary integral equations involving the operators

$$\begin{aligned} D_\Gamma \pm \textstyle {\frac{1}{2}} I \quad \text { and }\quad D'_\Gamma \pm \textstyle {\frac{1}{2}} I \end{aligned}$$

(1.1)

(see, e.g., [64, 46, Sects. 5.9, 5.15.1]), where the double-layer (or Neumann or Neumann-Poincaré) operator $D_{\Gamma }$ and the adjoint double-layer operator $D_{\Gamma }'$ are defined by

$$\begin{aligned} D_{\Gamma } \phi (x) = \int _\Gamma \frac{\partial \Phi (x,y)}{\partial n(y)} \phi (y)\, \textrm{d}s(y) \quad \text { and }\quad D_{\Gamma }' \phi (x) = \int _\Gamma \frac{\partial \Phi (x,y)}{\partial n(x)} \phi (y)\, \textrm{d}s(y), \end{aligned}$$

(1.2)

for $\phi \in {L^2(\Gamma )}$ and (almost all) $x \in \Gamma$, with the integrals understood, in general, as Cauchy principal values. Here $\Phi (x,y)$ is the fundamental solution for Laplace’s equation, defined by²

$$\begin{aligned} \Phi (x,y):= \frac{1}{2\pi } \log \left( \frac{1}{|x-y|}\right) , \;\; d= 2, \;\;\quad := \frac{1}{(d-2)c_d}\frac{1}{|x-y|^{d-2}}, \;\; d \ge 3, \end{aligned}$$

(1.3)

where $c_d$ is the surface measure of the unit sphere in $\mathbb {R}^d$. Explicitly,

$$\begin{aligned} D_{\Gamma } \phi (x)= & {} \frac{1}{c_d}\int _\Gamma \frac{(x-y) \cdot n(y)}{|x-y|^d} \phi (y)\, \textrm{d}s(y) \quad \text { and }\nonumber \\ D_{\Gamma }' \phi (x)= & {} \frac{1}{c_d}\int _\Gamma \frac{(y-x) \cdot n(x)}{|x-y|^d} \phi (y)\, \textrm{d}s(y), \end{aligned}$$

(1.4)

for $\phi \in {L^2(\Gamma )}$ and (almost all) $x \in \Gamma$.

Complementing (1.1),

$$\begin{aligned} D_{\Gamma }'-\lambda I, \end{aligned}$$

(1.5)

with $\lambda \in \mathbb {C}\setminus \{-\frac{1}{2},\frac{1}{2}\}$, arises as the operator in the boundary integral equation reformulation of transmission problems in electrostatics, where the Laplace equation $\Delta u = 0$ holds in $\Omega _-$ and $\Omega _+$ and the trace of u or its normal derivative jumps across $\Gamma$ (see, e.g., [46, Sect. 5.12]). In this context the case $|\lambda |\ge 1/2$, especially with $\lambda$ real, is classically of interest (e.g., [46, Sect. 5.12]); more recently the case where $\lambda$ is complex with $|\lambda |< 1/2$ has been studied intensively as a model of quasi-static electromagnetic plasmonic problems (e.g., [1, 15, 27, 59]).

Motivated by these physical applications, and by questions in harmonic analysis, there has been long-standing interest in the computation of the spectrum and essential spectrum³ of $D_\Gamma$ as an operator on a variety of function spaces, especially for non-smooth domains (e.g., [1, 12, 18, 27, 34, 38, 39, 48, 52, 53]). The largest part of this literature is concerned specifically with the 2D/3D cases where $\Gamma$ is a (curvilinear) polygon (e.g., [10, 49, 50, 60, 61]) or polyhedron (e.g., [15, 16, 20, 26, 27, 48, 50, 56]). In this paper we will study and compute the essential spectrum of $D_\Gamma$ as an operator on $L^2(\Gamma )$ for a substantially larger class of boundaries, namely for the case where the boundary $\Gamma$ is locally dilation invariant in the sense of Definition 1.3 below. In 2D (3D) this class includes polygons (polyhedra) but it also admits much wilder boundary behaviour (e.g., Fig. 1) as we discuss next in Sect. 1.1.

1.1 The spectral radius conjecture and the main question we address

Given a bounded linear operator $T:Y\rightarrow Y$ on a Banach space Y, we define its spectral radius, $\rho (T;Y)$, and its essential spectral radius, $\rho _{{{\,\textrm{ess}\,}}}(T;Y)$, by

$$\begin{aligned} \rho (T;Y) := \sup _{\lambda \in \sigma (T;Y)} |\lambda | \quad \text{ and } \quad \rho _{{{\,\textrm{ess}\,}}}(T;Y) := \sup _{\lambda \in \sigma _{{{\,\textrm{ess}\,}}}(T;Y)} |\lambda |, \end{aligned}$$

(1.6)

abbreviating $\rho (T;Y)$ and $\rho _{{{\,\textrm{ess}\,}}}(T;Y)$ by $\rho (T)$ and $\rho _{{{\,\textrm{ess}\,}}}(T)$, respectively, when the space Y is clear from the context. The analysis and computation we will carry out are motivated by the so-called spectral radius conjecture.

This conjecture, in the explicit 1994 formulation of Kenig [34], is as follows, where $L^2_0(\Gamma ):= \{\phi \in L^2(\Gamma ):\int _\Gamma \phi \,\textrm{d}s=0\}$.

Conjecture 1.1

If $\Gamma$ is the boundary of a bounded Lipschitz domain $\Omega _-$ and is connected, the spectral radius of $D_{\Gamma }'$ on $L^2_0(\Gamma )$ is $<\frac{1}{2}$, i.e. $\rho (D'_\Gamma ;L^2_0(\Gamma )) < \frac{1}{2}$.

In Sect. 2 we will discuss the following alternative formulation of the conjecture which makes sense regardless of the connectivity of $\Gamma$, and show its equivalence with Conjecture 1.1.

Conjecture 1.2

If $\Gamma$ is the boundary of a bounded Lipschitz domain $\Omega _-$, the essential spectral radius of $D_{\Gamma }$ on $L^2(\Gamma )$ is $<\frac{1}{2}$, i.e. $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma )) < \frac{1}{2}$.

Fig. 1

Examples of bounded Lipschitz domains $\Omega _-$ with boundaries $\Gamma \in \mathscr {D}_\mathcal {A}\subset \mathscr {D}$ that are piecewise analytic locally dilation invariant

The spectral radius conjecture is very well studied, owing to its intrinsic interest in harmonic analysis, its possible relevance for computation,⁴ and its immediate role within electrostatics, as well as in interpretations of electrodynamical problems [26]. It originated in the setting of continuous functions in $\overline{\Omega _{-}}$, dating all the way back to Neumann in the late 1800s, who treated convex domains, and to Radon [55], who famously analyzed curves of bounded rotation. In a tour de force, Král [38, 39] completely characterized when the essential norm of $D_\Gamma$ is $< 1/2$. This result was extended into higher dimensions by Burago and Maz’ya [8] and Netuka [51]. For polyhedra in 3D, the essential norm can be $> 1/2$. Nevertheless, Rathsfeld [56] (and see [57]) and Grachev and Maz’ya [20] independently proved the spectral radius conjecture in the continuous setting holds for general polyhedra. These results were extended to locally conformal deformations of polyhedra by Medková [45]. Even when specialised to the continuous setting, the history is vast, and we refer to Wendland [65] for an in-depth survey.

As modern harmonic analysis developed, the natural setting for the double-layer potential shifted toward $L^2(\Gamma )$. We make particular mention of the demonstrations of $L^2$-boundedness of the Cauchy integral due to Calderón [9] and Coifman, Mcintosh, and Meyer [13], and Verchota’s [64] application of these results to study invertibility of $D'_\Gamma \pm \frac{1}{2}I$ on $L^2(\Gamma )$ and $L^2_0(\Gamma )$ when $\Gamma$ is connected. Since then, a flurry of activity and findings in this area have provided support for Kenig’s conjecture, though a complete proof has proved elusive.

Indeed, to the best of our knowledge, Conjecture 1.1 has been established (only) in the following cases: (a) $\Omega _-$ is convex [18] (and see [12] for extensions to locally convex domains); (b) $\Omega _-$ has small Lipschitz character⁵ [48], a case which includes all $C^1$ domains [19]; (c) $\varepsilon$-regular Semmes–Kenig–Toro domains $\Omega _-$ for sufficiently small $\varepsilon > 0$ [30], including in particular all domains whose gradient has vanishing mean oscillation [28]; (d) $\Omega _-$ is a polygon or curvilinear polygon in 2D [60, 61], or a Lipschitz polyhedron in 3D [16].

Note that polygonal and polyhedral boundaries $\Gamma$ are locally invariant under all dilations: at every point $x\in \Gamma$, $\Gamma$ locally coincides with a graph $\Gamma _x$ such that (in some local coordinate system centred at x) $\Gamma _x = \alpha \Gamma _x$ for all $\alpha >0$. In this paper we will investigate domains where, locally, the dilation invariance $\Gamma _x = \alpha _x \Gamma _x$ only holds for one $\alpha _x \in (0,1)$ (in which case we say that $\Gamma$ is locally dilation invariant at x). Precisely our focus will be on the following class of boundaries.

Definition 1.3

(Locally dilation invariant) Given $\Gamma$, the boundary of a Lipschitz domain $\Omega _-$, we say that $\Gamma \in \mathscr {D}$, the set of locally-dilation-invariant boundaries if, at every $x \in \Gamma$, $\Gamma$ is either locally $C^1$ or locally coincides (in some coordinate system centred at x) with a Lipschitz graph $\Gamma _x$ which is dilation invariant with respect to some $\alpha _x \in (0,1)$, i.e. $\Gamma _x = \alpha _x \Gamma _x$.

Note that, already in 2D, $\mathscr {D}$ is a hugely larger class of domains than that of the curvilinear polygons. Indeed, for a Jordan curve $\Gamma \subset \mathbb {R}^2$ and $x \in \Gamma$, let

$$\begin{aligned} v_\Gamma (x) = \int _{0}^{2\pi } |\{x + re^{i\theta } \in \Gamma \,: \, r > 0 \}| \, \frac{d\theta }{2\pi }, \end{aligned}$$

where $\left|\cdot \right|$ in this equation denotes the counting measure of a set. Then $D_\Gamma \phi$ is uniformly continuous on $\Omega _-$ for every $\phi \in C(\Gamma )$ if and only if

$$\begin{aligned} \sup _{x\in \Gamma } v_\Gamma (x) < \infty , \end{aligned}$$

(1.7)

see [7, 36, 37]. If $\Gamma$ is Lipschitz and locally dilation invariant at x, then it is clear that $v_\Gamma (x) = \infty$, unless $\Gamma$ coincides with two line segments around x. It follows that the only curves $\Gamma \in \mathscr {D}$ satisfying (1.7) are curvilinear polygons. That is, except for curvilinear polygons, our curves exhibit such wild boundary behaviour that it is not possible to consider $D_\Gamma$ in the setting of continuous boundary data.

In the context of studying the double-layer and related operators, the class of domains $\mathscr {D}$ seems to have been first considered in [11], where the essential numerical range⁶ of $D_\Gamma$ was studied and, in 2D and 3D, examples of boundaries $\Gamma \in \mathscr {D}$ with large Lipschitz character were constructed such that $D_\Gamma :L^2(\Gamma ) \rightarrow L^2(\Gamma )$ has arbitrarily large essential numerical radius,

$$\begin{aligned} w_{{{\,\textrm{ess}\,}}}(D_\Gamma ):= \sup \limits _{z \in W_{{{\,\textrm{ess}\,}}}(D_\Gamma )} |z|, \end{aligned}$$

and so also arbitrarily large essential norm,

$$\begin{aligned} \Vert D_\Gamma \Vert _{{{\,\textrm{ess}\,}}}:= \inf _{K \text{ compact }}\Vert D_\Gamma -K\Vert , \end{aligned}$$

since, for any bounded operator T on a Hilbert space [22, Sect. 1.3],

$$\begin{aligned} \textstyle {\frac{1}{2}} \Vert T\Vert _{{{\,\textrm{ess}\,}}} \le w_{{{\,\textrm{ess}\,}}}(T) \le \Vert T\Vert _{{{\,\textrm{ess}\,}}}. \end{aligned}$$

(1.8)

The 2D examples in [11] (see, e.g., [11, Fig. 3] and cf. Fig. 1) with arbitrarily large $w_{{{\,\textrm{ess}\,}}}(D_\Gamma )$ are necessarily examples for which (1.7) fails to hold, since for curvilinear polygons on $L^2(\Gamma )$ it is well-known that

$$\begin{aligned} \Vert D_\Gamma \Vert _{{{\,\textrm{ess}\,}}}= w_{{{\,\textrm{ess}\,}}}(D_\Gamma )=\rho (D_\Gamma ;L^2(\Gamma ))<\textstyle {\frac{1}{2}} \end{aligned}$$

(see [10, 49, 60, 61] for $\Vert D_\Gamma \Vert _{{{\,\textrm{ess}\,}}}$ and $\rho (D_\Gamma ;L^2(\Gamma ))$; equality for $w_{{{\,\textrm{ess}\,}}}(D_\Gamma )$ follows by (1.8) and since $W_{{{\,\textrm{ess}\,}}}(D_\Gamma )\supset \sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma ))$ [6]).

A natural question, prompted by the examples from Chandler-Wilde and Spence [11] with $w_{{{\,\textrm{ess}\,}}}(D_\Gamma )$ arbitrarily large and the spectral radius conjecture, is the following:

Given that there exist $\Gamma \in \mathscr {D}$ with $w_{{{\,\textrm{ess}\,}}}(D_\Gamma )\gg \frac{1}{2}$, in particular such examples in 2D, is there a $\Gamma \in \mathscr {D}$, in particular an example in 2D, with $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma ))\ge \frac{1}{2}$?

Of course, a positive answer would provide a counterexample to Conjecture 1.2 and hence to the original spectral radius conjecture, Conjecture 1.1. The aim of this paper is to address this question, through mathematical analysis and computational methods supported by numerical analysis error estimates where constants are made explicit. These will enable us to estimate $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma ))$ for a large class of $\Gamma \in \mathscr {D}$ sufficiently accurately to determine whether or not $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma ))\ge \frac{1}{2}$.

1.2 Our main results and their significance

The first step in our analysis is Theorem 5.2 in Sect. 5.1, the localisation result (cf. [11, 15, 16, 48]) that, for $\Gamma \in \mathscr {D}$ (and for any dimension $d\ge 2$), there exists a finite set $F\subset \Gamma$ such that

$$\begin{aligned} \sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma )) = \bigcup _{x\in F} \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _x};L^2({\Gamma _x})). \end{aligned}$$

(1.9)

In the above formula $\Gamma _x$ is defined as above if $\Gamma$ is locally dilation invariant at x, while if $\Gamma$ is locally $C^1$ at x then $\Gamma _x$ is the graph of a $C^1$ compactly supported function so that [19] $D_{\Gamma _x}$ is compact and $\sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _x};L^2({\Gamma _x}))=\{0\}$.

The localisation (1.9) reduces the computation of $\sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma ))$ to that of $\sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _x};L^2({\Gamma _x}))$ for finitely many x for which $\alpha _x\Gamma _x = \Gamma _x$, for some $\alpha _x\in (0,1)$. Computation of $\sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _x};L^2({\Gamma _x}))$ for such x is our focus in Sect. 4 where we study and compute spectral properties of $D_\Gamma$ in the case that $\Gamma \subset {\mathbb {R}}^d$ is a dilation invariant Lipschitz graph, meaning that there exists an $\alpha \in (0,1)$ such that $\alpha \Gamma = \Gamma$ (an example is Fig. 3). If $\Gamma _0\subset \Gamma \setminus \{0\}$ is a particular relatively closed and bounded Lipschitz subgraph of $\Gamma$ such that

$$\begin{aligned} \Gamma = \bigcup _{j \in {\mathbb {Z}}} \alpha ^j \Gamma _0, \end{aligned}$$

(1.10)

and such that $(\alpha ^j\Gamma _0)\cap \Gamma _0$ has zero surface measure for $j\ne 0$, we show that $D_\Gamma$ can be written as a discrete $\ell _1$ convolution operator whose entries are bounded linear operators on $L^2(\Gamma _0)$ related to the discretization (1.10) of $\Gamma$. This allows us to decompose $D_\Gamma$, by a Floquet–Bloch transform, into a continuous family $(K_t)_{t \in [-\pi , \pi ]}$ of operators $K_t :L^2(\Gamma _0) \rightarrow L^2(\Gamma _0)$, and, by standard results for such convolutions (see [54, Theorem 2.3.25]), to characterise the essential spectrum of $D_\Gamma$ as (Theorem 4.4)

$$\begin{aligned} \sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma )) = \sigma (D_\Gamma ;L^2(\Gamma )) = \bigcup _{t\in [-\pi ,\pi ]} \sigma (K_t;L^2(\Gamma _0)). \end{aligned}$$

(1.11)

This characterisation is particularly useful when, apart from a singularity at 0, $\Gamma$ is $C^1$ (see, e.g., Fig. 3), for then (Corollary 4.6) each $K_t$ is compact, and so has a more easily computed discrete spectrum. The characterisation (1.11) holds for every Lipschitz dilation invariant graph in any dimension; in Sect. 4.3 we specialise to the 2D case where $\Gamma$, except at 0, is the graph of a real analytic function (see, e.g., Fig. 3); we denote by $\mathcal {A}$ this subset of the 2D dilation invariant graphs. We show, for each $t\in [-\pi ,\pi ]$, that $K_t$ is unitarily equivalent to ${\tilde{K}}_t$, a $2\times 2$ matrix of integral operators on $L^2(0,1)$ that have real analytic kernels. Further, the range of each of these integral operators is a space of 1-quasi-periodic real analytic functions. As a consequence,

$$\begin{aligned} \sigma (K_t;L^2(\Gamma _0)) = \sigma ({\tilde{K}}_t;(C[0,1])^2), \end{aligned}$$

and this latter spectrum can be computed by approximating ${\tilde{K}}_t$ by a $2N\times 2N$ matrix ${\tilde{A}}_{t,N}$ obtained by a simple midpoint-rule based Nyström discretization. As is well-known (see [41, 63], Theorem 4.11), the midpoint rule is exponentially convergent for periodic analytic functions, so that it follows from Nyström-method spectral estimates for integral operators with continuous kernels [3] that, for each t, the eigenvalues of $A_{t,N}$ converge at an exponential rate to those of ${\tilde{K}}_t$ as $N\rightarrow \infty$.

This leads (see Theorem 4.19) to a Nyström approximation, $\sigma ^N(D_\Gamma )$, for $\sigma (D_\Gamma )=\sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma )$, which is $\{0\}$ plus the union of the eigenvalues of finitely many $2N\times 2N$ matrices. Our first main result is to show, as Theorem 4.19, that $\sigma ^N(D_\Gamma )\rightarrow \sigma (D_\Gamma )$ in the Hausdorff metric as $N\rightarrow \infty$; that this convergence is achievable is somewhat surprising given that $D_\Gamma$ is neither compact nor self-adjoint. Our second, and more substantial result (Theorem 4.21) is that we develop a fully discrete algorithm to test whether, as an operator on $L^2(\Gamma )$, $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$. Precisely, we construct (see Remark 4.22), for each $c>0$, a nonlinear functional $S_{c}:\mathcal {A}\times \mathbb {N}\times \mathbb {N}\times \mathbb {N}\rightarrow \mathbb {R}$ with the properties that: a) the functional can be computed in finitely many arithmetic operations and finitely many evaluations of elementary functions, given finitely many sampled values of f and its first and second derivatives for real arguments, plus bounds on the analytic continuation of f to a neighbourhood of the real line that depends on c; b) $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$ if, for some $c>0$ and $m,M,N\in \mathbb {N}$, it holds that:

• (i) $\rho (A^M_{t_k,N})< \frac{1}{2}$, for $k=1,\ldots ,m$, where $t_k:= (k-1/2)\pi /m$ and $A^M_{t_k,N}$ is a specific approximation to $A_{t_k,N}$ depending on the parameter M;

• (ii) $S_{c}(\Gamma ,m,M,N) < 0$.

Conversely, if $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$, then, for all sufficiently small $c>0$, (i) and (ii) hold for all sufficiently large $N,M,m\in \mathbb {N}$.

In Sect. 6 we bring these results together to address our question at the end of Sect. 1.1. We restrict attention to the following 2D class of domains; this class includes polygons and piecewise-analytic curvilinear polygons, but also wilder boundary behaviour as illustrated in Fig. 1.

Definition 1.4

(Piecewise analytic locally dilation invariant) In the case $d=2$ we say that $\Gamma \in \mathscr {D}_\mathcal {A}$, the set of piecewise-analytic locally-dilation-invariant boundaries, if $\Gamma \in \mathscr {D}$ and $\Gamma$ is locally analytic (i.e., is locally the graph of a real-analytic function) at all but finitely many $x\in \Gamma$.

Given $\Gamma \in \mathscr {D}_\mathcal {A}$, if $F\subset \Gamma$ is the finite set of points at which $\Gamma$ is not locally analytic and is locally dilation invariant, it is easy to see that $\Gamma _x\in \mathcal {A}$ for $x\in F$ and that (1.9) holds for this set F. Thus (see Theorem 6.1)

$$\begin{aligned} \Sigma ^N(D_\Gamma ) := \bigcup _{x\in F} \sigma ^N(D_{\Gamma _x}) \rightarrow \sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma ) \end{aligned}$$

(1.12)

in the Hausdorff metric as $N\rightarrow \infty$, and note that $\Sigma ^N(D_\Gamma )$ is the union of the eigenvalues of finitely many $2N\times 2N$ matrices. Further, as we discuss in Sect. 6, it follows that $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$ if these eigenvalues lie within the disc of radius $\frac{1}{2}$, i.e. if

$$\begin{aligned} R_N(D_\Gamma ):=\max \{|z|: z\in \Sigma ^N(D_\Gamma )\} < \textstyle {\frac{1}{2}}, \end{aligned}$$

(1.13)

and if also, for some $c>0$,

$$\begin{aligned} \mathscr {S}_{c}(\Gamma , N) := \max _{x\in F} S_{c}(\Gamma _x,N,N,N) < 0, \end{aligned}$$

(1.14)

where $\mathscr {S}_{c}(\Gamma , N)$ can be computed in finitely many arithmetic operations plus finitely many evaluations of elementary functions, given inputs describing each $\Gamma _x$ as discussed above. Conversely, if $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$ then, for all sufficiently small $c>0$ and all sufficiently large N, $R_N(D_\Gamma )<\frac{1}{2}$ and $\mathscr {S}_{c}(\Gamma , N)<0$. Thus, given inputs describing $\Gamma$, our fully discrete algorithms enable us to test, for individual $\Gamma \in \mathscr {D}_\mathcal {A}$, the validity of Conjecture 1.2, i.e. whether or not $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$, through computation of the eigenvalues of finitely many finite matrices, plus finitely many additional arithmetic operations.

In Sect. 5 we prove the localisation result (1.9). We also, in the spirit of Medková’s study in the continuous setting [43, 45], consider the stability of Conjecture 1.2 under locally conformal deformations. We prove that the spectral radius conjecture is independent of such deformations, under the assumption that the absolute value of the kernel of the double-layer potential also defines a bounded operator on $L^2(\Gamma )$. While this additional hypothesis may fail to include the wildest of boundaries, it applies to many domains from $\mathscr {D}$. For example, as we discuss in Sect. 5.2, it is satisfied by any dilation invariant Lipschitz graph $\Gamma$ whose generating set $\Gamma _0$ is polygonal or polyhedral, in particular it holds if $\Gamma$ is a polyhedron and, in 2D, if $\Gamma \in \mathscr {D}_\mathcal {A}$. As one consequence (Corollary 5.7), Conjecture 1.2 holds for Lipschitz curvilinear polyhedra, because [16] it holds for polyhedra; as another (Corollary 5.11), if it holds for $\Gamma \in \mathscr {D}_\mathcal {A}$, then it holds for any locally conformal deformation of $\Gamma$.

To illustrate the above results, and use them to test Conjecture 1.2 and address our main question from Sect. 1.1, we include a range of numerical examples in Sect. 6, for piecewise-analytic Lipschitz graphs $\Gamma \in \mathcal {A}$, and for piecewise-analytic locally-dilation-invariant $\Gamma \in \mathscr {D}_\mathcal {A}$ that are the boundaries of bounded Lipschitz domains. For each example we plot an approximation ($\sigma ^N(D_\Gamma )$ or $\Sigma ^N(D_\Gamma )$, as appropriate) to $\sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma )$. Moreover, we employ the algorithms described above to provide convincing numerical evidence that $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma ))<\frac{1}{2}$ in every case we examine, including cases where $w_{{{\,\textrm{ess}\,}}}(D_\Gamma )$ is significantly $>\frac{1}{2}$. These results are evidence that the spectral radius conjecture holds for the class of 2D domains $\mathscr {D}_\mathcal {A}$; we emphasise again that this conjecture has not been studied previously for boundaries in this class, except for the special case of curvilinear polygons.

Let us briefly summarise the remainder of the paper. In Sect. 2 we prove the equivalence of Conjectures 1.1 and 1.2. In Sect. 3, as a key step to our main results, we derive bounds for the spectral radii of general compact operators in Sect. 3.1 (e.g., Corollary 3.5), specialising to the case of integral operators approximated by the Nyström method in Sect. 3.2 (e.g., Theorem 3.8). In Sect. 4 we prove the results noted above on the essential spectrum and essential spectral radius of $D_\Gamma$ in the case when $\Gamma$ is a dilation-invariant Lipschitz graph, with particular focus (Sects. 4.2–4.4) on the 2D piecewise-analytic case. In Sect. 5 we prove our localisation results. In Sect. 6 we bring the earlier results together, in particular to study the spectral radius conjecture for $\Gamma \in \mathscr {D}_\mathcal {A}$, and we illustrate our theory by numerical examples.

2 Formulations of the spectral radius conjecture

Our results are related to the spectral radius conjecture of Kenig [34, Problem 3.2.12], that $\rho (D_{\Gamma }';L^2_0(\Gamma ))<\frac{1}{2}$ if $\Gamma$ is the boundary of a bounded Lipschitz domain. Conjecture 1.1, stated in the introduction, is a version of this conjecture that avoids difficulties when $\Gamma$ is not connected⁷. In this section we show the equivalence between Conjecture 1.1 and Conjecture 1.2, also stated in the introduction. Conjecture 1.2 concerns the essential spectrum rather than the spectrum and makes sense whatever the connectedness of $\Gamma$, $\Omega _-$, and $\Omega _+$.

Of course, because $D_{\Gamma }'$ is the adjoint of $D_{\Gamma }$ (as an operator on $L^2(\Gamma )$), and so shares the same essential spectrum, a statement equivalent to Conjecture 1.2 is that $\rho _{{{\,\textrm{ess}\,}}}(D_{\Gamma }';L^2(\Gamma ))<\frac{1}{2}$. Further, as bounded Lipschitz domains have only finitely many boundary components, it is clear that Conjecture 1.2 is true if it is true whenever $\Gamma$ is connected.⁸ Thus the equivalence of Conjectures 1.1 and 1.2, i.e. that Conjecture 1.1 holds (as claimed, whenever $\Gamma$ is connected) if and only if Conjecture 1.2 holds (as claimed, whatever the topology of $\Gamma$), is implied by the following lemma.

Lemma 2.1

Assume that $\Gamma$ is the boundary of a bounded Lipschitz domain $\Omega _-$ and is connected. Then $\rho (D_{\Gamma }';L^2_0(\Gamma ))<\frac{1}{2}$ if and only if $\rho _{{{\,\textrm{ess}\,}}}(D_{\Gamma }';L^2(\Gamma ))<\frac{1}{2}$.

The main step in the proof of this lemma is the following theorem stated in [18].⁹ As noted in [18], the proof of this result is that in Kellogg’s classical book for the case of smooth boundaries (see [33, Chapter IX, Sect. 11]), which carries over to the Lipschitz case.

Theorem 2.2

[Theorem 1.1 in [18]] If $\Gamma$ is the boundary of a bounded Lipschitz domain $\Omega _-$, the eigenvalues of $D_{\Gamma }'$, as an operator on $L^2(\Gamma )$, are real and lie in $[-\frac{1}{2},\frac{1}{2}]$.

Proof of Lemma 2.1

Let $\phi \in L^2(\Gamma )$ and let $\langle \cdot ,\cdot \rangle$ denote the inner product on $L^2(\Gamma )$. We first observe that, since $D_{\Gamma }(1)=-\frac{1}{2}$, we have

$$\begin{aligned} \langle D_{\Gamma }'\phi ,1\rangle = \langle \phi ,D_{\Gamma }(1)\rangle = -\tfrac{1}{2} \langle \phi ,1\rangle , \end{aligned}$$

(2.1)

so $D_{\Gamma }'(L^2_0(\Gamma )) \subseteq L^2_0(\Gamma )$.

Now assume that $\lambda \in \mathbb {C}$ and $D_{\Gamma }' - \lambda I$ is invertible as an operator on $L^2_0(\Gamma )$. Let P be orthogonal projection from $L^2(\Gamma )$ onto the constants, so that $Q:=I-P$ is projection onto $L^2_0(\Gamma )$. Then $(D_{\Gamma }' - \lambda )Q+P$ is invertible as an operator on $L^2(\Gamma )$. It follows that

$$\begin{aligned} D_{\Gamma }' - \lambda I = ((D_{\Gamma }' - \lambda )Q+P) + ((D_{\Gamma }' - \lambda )P-P) \end{aligned}$$

(2.2)

is Fredholm as an operator on $L^2(\Gamma )$. This implies that $\sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma }';L^2(\Gamma )) \subset \sigma (D_{\Gamma }';L^2_0(\Gamma ))$, which settles one direction.

Conversely, assume that $\rho _{{{\,\textrm{ess}\,}}}(D_{\Gamma }';L^2(\Gamma ))<\frac{1}{2}$. Let $\lambda \in \mathbb {C}$ with $|\lambda | \ge \frac{1}{2}$. Then $D_{\Gamma }' - \lambda I$ is Fredholm of index 0 on $L^2(\Gamma )$, so [see (2.2)] $(D_{\Gamma }' - \lambda )Q +P$ is Fredholm of index zero on $L^2(\Gamma )$, so that $D_{\Gamma }' - \lambda I$ is Fredholm of index zero on $L^2_0(\Gamma )$. Thus $\lambda \in \sigma (D_{\Gamma }';L^2_0(\Gamma ))$ if and only if $\lambda$ is an eigenvalue of $D_{\Gamma }'$. Hence, if $\lambda \notin [-\frac{1}{2},\frac{1}{2}]$, $D_{\Gamma }' - \lambda I$ is invertible on both $L^2(\Gamma )$ and $L^2_0(\Gamma )$ by Theorem 2.2 (as $L^2_0(\Gamma ) \subseteq L^2(\Gamma )$, every eigenvalue on $L^2_0(\Gamma )$ is also an eigenvalue on $L^2(\Gamma )$). But also, if $\lambda = \pm \frac{1}{2}$, since $\Gamma$ is connected, Verchota’s results [64] show that $D_{\Gamma }' \pm \frac{1}{2}I$ is invertible on $L^2_0(\Gamma )$. Thus $D_{\Gamma }' - \lambda I$ is invertible on $L^2_0(\Gamma )$ for $\left|\lambda \right| \ge \frac{1}{2}$, so that, since the spectrum is closed, $\rho (D_{\Gamma }';L^2_0(\Gamma ))<\frac{1}{2}$. $\square$

3 Approximation of the spectral radius for compact operators

In this section we recall in Sect. 3.1 results from operator approximation theory in Banach spaces related to the spectra of compact operators, and derive what appear to be new general criteria for $\rho (T)<\rho _0$ when T is compact and $\rho _0>0$ (Lemma 3.2, Corollary 3.5). This leads, in Sect. 3.2, to results relating to the approximation of integral operators with continuous kernels by the Nyström method that will be key for the arguments in Sect. 4.2 and Sect. 4.3. Notably, Theorem 3.8 provides criteria for $\rho (T)<\rho _0$ when T is an integral operator with a continuous kernel that requires the computation only of the spectral radius of a finite matrix plus the norms of finitely many finite matrix resolvents.

3.1 Operator approximation results

We recall first two standard results on the approximation of operators in $\mathcal {L}(Y)$, the space of bounded linear operators on a Banach space Y. The first is the basic perturbation estimate that, if $S,T\in \mathcal {L}(Y)$ and S is invertible, then T is invertible if $\left\Vert S-T\right\Vert \left\Vert S^{-1}\right\Vert < 1$, with

$$\begin{aligned} \left\Vert T^{-1}\right\Vert ^{-1} \ge \left\Vert S^{-1}\right\Vert ^{-1} - \left\Vert S-T\right\Vert . \end{aligned}$$

(3.1)

If $S\in \mathcal {L}(Y)$ is invertible, then $\Vert S^{-1}\Vert ^{-1}=\inf _{\Vert \phi \Vert = 1} \Vert S\phi \Vert$, this sometimes called the lower norm of S (see, e.g., [42, Lemma 2.35]). The second estimate is as follows:

Lemma 3.1

(Theorem 4.7.7 of [23]) Let Y be a Banach space, $S \in \mathcal {L}(Y)$ and ${\lambda } \notin \sigma (S) \cup \{0\}$. If $T \in \mathcal {L}(Y)$ is a compact operator that satisfies

$$\begin{aligned} \left\Vert (T-S)T\right\Vert < \left|\lambda \right|\left\Vert (S-\lambda I)^{-1}\right\Vert ^{-1}, \end{aligned}$$

then $T - \lambda I$ is invertible.

The following result is a consequence of the above estimates and the maximum principle applied to the resolvent. Here $\mathbb {T}=\{z:|z|=1\}$ is the unit circle in the complex plane, so that $\rho _0\mathbb {T}$ is the circle of radius $\rho _0$.

Lemma 3.2

Let Y be a Banach space, $S,\widehat{S},T \in \mathcal {L}(Y)$, $\rho _0>0$, $F\subset \rho _0\mathbb {T}$, and suppose that: T is compact; $\rho (\widehat{S})<\rho _0$; for every $\lambda \in \rho _0\mathbb {T}$ there exists $\mu \in F$ such that

$$\begin{aligned} \left\Vert (T-S)T\right\Vert < \rho _0\left( \left\Vert (\widehat{S}-\mu I)^{-1}\right\Vert ^{-1} - \left\Vert S-\widehat{S}\right\Vert - |\lambda -\mu |\right) . \end{aligned}$$

(3.2)

Then $\rho (T)<\rho _0$.

The idea is to choose F to be a finite set,¹⁰S a finite rank approximation to T, and $\widehat{S}$ a numerical approximation to S, in which case one can show $\rho (T)<\rho _0$ by computing $\rho (\widehat{S})$ and $\Vert (\widehat{S}-\mu I)^{-1}\Vert$ for finitely many $\mu$. Taking $F=\rho _0\mathbb {T}_n$, where $\mathbb {T}_n$ is the nth roots of unity, we obtain:

Corollary 3.3

Let Y be a Banach space, $S,\widehat{S},T \in \mathcal {L}(Y)$, $\rho _0>0$, $n\in \mathbb {N}$, and suppose that T is compact, $\rho (\widehat{S})<\rho _0$, and

$$\begin{aligned} \left\Vert (T-S)T\right\Vert < \rho _0\left( \Vert (\widehat{S}-\mu I)^{-1}\Vert ^{-1} - \Vert S-\widehat{S}\Vert - 2\rho _0\sin \left( \frac{\pi }{2n}\right) \right) , \qquad \mu \in \rho _0 \mathbb {T}_n. \end{aligned}$$

(3.3)

Then $\rho (T)<\rho _0$.

Proof of Lemma 3.2

Suppose that the conditions of the lemma are satisfied. Then $\rho (\widehat{S})< \rho _0$ and, for every $\lambda \in \rho _0\mathbb {T}$, there exists $\mu \in F$ such that (3.2) holds. It follows from (3.1) that

$$\begin{aligned} \left\Vert (T-S)T\right\Vert < \rho _0\left( \Vert (\widehat{S}-\lambda I)^{-1}\Vert ^{-1} - \Vert S-\widehat{S}\Vert \right) , \quad \lambda \in \rho _0\mathbb {T}. \end{aligned}$$

(3.4)

The resolvent map $\lambda \mapsto (\widehat{S}-\lambda I)^{-1}$ is analytic on $|\lambda |>\rho (\widehat{S})$, which set contains all $\lambda$ with $|\lambda |\ge \rho _0$. Thus, by the maximum principle, $\Vert (\widehat{S}-\lambda I)^{-1}\Vert$ attains its maximum in $|\lambda | \ge \rho _0$ on $\rho _0\mathbb {T}$. Thus (3.4) in fact holds for all $\lambda$ with $|\lambda |\ge \rho _0$, so that $S-\lambda I$ is invertible for all such $\lambda$ and, by (3.1),

$$\begin{aligned} \left\Vert (T-S)T\right\Vert < \rho _0 \left\Vert (S-\lambda I)^{-1}\right\Vert ^{-1}, \quad \text{ if } |\lambda | \ge \rho _0. \end{aligned}$$

(3.5)

Since T is compact, the result follows from Lemma 3.1. $\square$

When Lemma 3.2 is used for computation with F finite, it is desirable to minimise the cardinality of F since $\Vert (\widehat{S}-\mu I)^{-1}\Vert$ has to be computed for every $\mu \in F$. One can choose $F=\rho _0\mathbb {T}_n$, with points uniformly distributed on $\rho _0\mathbb {T}$, as in Corollary 3.3, but n needs to be at least large enough so that $\Vert (\widehat{S}-\mu I)^{-1}\Vert ^{-1} > 2\rho _0\sin (\pi /(2n))$, for every $\mu \in F$. In many applications, including in Sects. 4.2 and 4.3, $\Vert (\widehat{S}-\mu I)^{-1}\Vert$ varies significantly as $\mu$ moves around $\rho _0\mathbb {T}$ and it is more efficient to vary the spacing of the points in F approximately in proportion to $\Vert (\widehat{S}-\mu I)^{-1}\Vert ^{-1}$. The adaptive algorithm described in the following lemma, which we will see implemented in Fig. 4b below, approximately achieves this.

Lemma 3.4

Let Y be a Banach space, $\widehat{S}\in \mathcal {L}(Y)$, and $\rho _0>0$. Suppose that $\rho (\widehat{S})< \rho _0$, and recursively define $\mu _\ell$, for $\ell \in \mathbb {N}$, by $\mu _{1}:= \rho _0$, and by

$$\begin{aligned} \nu _{\ell } := \left\Vert (\widehat{S} - \mu _{\ell } I)^{-1}\right\Vert ^{-1} \quad \textrm{and} \quad \mu _{\ell +1} := \mu _{\ell }e^{i\frac{\nu _{\ell }}{2\rho _0}}, \quad \textrm{for } \,\ell \in \mathbb {N}. \end{aligned}$$

(3.6)

Further, set $n_* \in \mathbb {N}$ to be the smallest integer such that $\sum _{\ell = 1}^{n_*} \nu _{\ell }\ge 4\pi \rho _0$, and set $F:= \{\mu _1,\mu _2,\ldots ,\mu _{n_*+1}\}$. Then, for every $\lambda \in \rho _0\mathbb {T}$ there exists $\mu \in F$ such that

$$\begin{aligned} \left\Vert (\widehat{S} - \lambda I)^{-1}\right\Vert ^{-1} \ge \left\Vert (\widehat{S} - \mu I)^{-1}\right\Vert ^{-1}-|\lambda -\mu | \ge R_* := \min _{\ell =1,\ldots ,n_*} \left( \frac{\nu _\ell }{4}+\frac{\nu _{\ell +1}}{2}\right) . \end{aligned}$$

(3.7)

Proof

Arguing as in the proof of Lemma 3.2, $\Vert (\widehat{S} -\lambda I)^{-1}\Vert ^{-1}$ is bounded below by some $c>0$ on $\rho _0\mathbb {T}$, so that $n_*$ is well-defined with $n_*<1+4\pi \rho _0/c$. For $\ell =1,\ldots ,n_*+1$, $\mu _\ell = \rho _0e^{i\theta _\ell }$, with $\theta _1=0$, $\theta _{n_*+1}\ge 2\pi$, and $\theta _{\ell +1}-\theta _\ell = \nu _\ell /(2\rho _0)$. Thus, if $\lambda \in \rho _0\mathbb {T}$, then

$$\begin{aligned} \lambda = \mu _\ell \exp (is\nu _\ell /(2\rho _0)) = \mu _{\ell +1}\exp (i(s-1)\nu _\ell /(2\rho _0)), \end{aligned}$$

for some $\ell \in \{1,\ldots ,n_*\}$ and some $s\in [0,1]$. Since $|e^{it}-1|=|\int _0^t e^{iu}\, \textrm{d}u|\le |t|$, for $t\in \mathbb {R}$, it follows, using (3.1), that

$$\begin{aligned} \Vert (\widehat{S} - \lambda I)^{-1}\Vert ^{-1}\ge & {} \nu _\ell - |\lambda - \mu _\ell | \ge \nu _\ell - \frac{s}{2}\nu _\ell = \left( 1-\frac{s}{2}\right) \nu _\ell \quad \text{ and }\\ \Vert (\widehat{S} - \lambda I)^{-1}\Vert ^{-1}\ge & {} \nu _{\ell +1} - |\lambda - \mu _{\ell +1}| \ge \nu _{\ell +1} - \frac{1-s}{2}\nu _\ell . \end{aligned}$$

In particular, $\nu _{\ell +1} \ge \nu _\ell /2$ and $\nu _\ell \ge \nu _{\ell +1} - \nu _\ell /2$, so that $s^*:= 3/2-\nu _{\ell +1}/\nu _\ell \in [0,1]$. But, for $0\le s\le s^{*}$, $(1-s/2)\nu _\ell \ge (1-s^*/2)\nu _\ell =\nu _\ell /4+\nu _{\ell +1}/2$, while, for $s^*\le s\le 1$, $\nu _{\ell +1} - (1-s)\nu _\ell /2 \ge \nu _{\ell +1} - (1-s^*)\nu _\ell /2 = \nu _\ell /4+\nu _{\ell +1}/2$, and the bound (3.7) follows. $\square$

The following corollary is immediate from the above lemma and Lemma 3.2.

Corollary 3.5

Let Y be a Banach space, $S,\widehat{S},T \in \mathcal {L}(Y)$, $\rho _0>0$, and suppose that T is compact, $\rho (\widehat{S})<\rho _0$, and

$$\begin{aligned} \left\Vert (T-S)T\right\Vert < \rho _0\left( R_* - \Vert S-\widehat{S}\Vert \right) , \end{aligned}$$

(3.8)

where $R_*$ is as defined in Lemma 3.4. Then $\rho (T)<\rho _0$.

We will apply the above results in the case when $S=T_N$, where $(T_N)_{N\in \mathbb {N}}$ is a collectively compact¹¹ sequence of operators converging strongly to T (we write $T_N\rightarrow T$ for strong convergence). A standard, simple but important result (e.g., [2, Cor. 1.9], [41, Theorem 10.10]) is that

$$\begin{aligned} (T_N)_{N\in \mathbb {N}} \text{ collectively } \text{ compact }, \quad T_N\rightarrow T \quad \Rightarrow \quad \Vert (T-T_N)T\Vert \rightarrow 0. \end{aligned}$$

(3.9)

A consequence of (3.9) is Theorem 3.6 below, which follows from [2, Theorem 4.8] and [2, Theorem 4.16] (or see [3]). This gives conditions on operators $T_N$ and T that ensure convergence of $\sigma (T_N)$ to $\sigma (T)$ in the standard Hausdorff metric $d_H(\cdot ,\cdot )$ (see, e.g., [24, Sect. 3.1.2]) on the set $\mathbb {C}^C$ of compact subsets of $\mathbb {C}$. Given a sequence $(A_N)_{N\in \mathbb {N}}\subset \mathbb {C}^C$ and $A\in \mathbb {C}^C$ we will write if $d_H(A,B)\rightarrow 0$, i.e. if $A_N$ converges to A in the Hausdorff metric. We recall (e.g., [24, Proposition 3.6]) that if and only if $(A_N)_{N\in \mathbb {N}}$ is uniformly bounded and $\liminf A_N = \limsup A_N = A$, where $\liminf A_N$ is the set of limits of sequences $(z_N)$ such that $z_N\in A_N$ for each N, while $\limsup A_N$ is the set of partial limits of such sequences.

Theorem 3.6

[2] Let Y be a Banach space and $T\in \mathcal {L}(Y)$ a compact operator. Further, assume that $(T_N)_{N \in \mathbb {N}}\subset \mathcal {L}(Y)$ is collectively compact and converging strongly to T. Then .

3.2 The Nyström method

In this section, with a view to applications in Sects. 4.2 and 4.3, we apply the results of Sect. 3.1 to the case where $Y=C(X)$, for some compact $X\subset \mathbb {R}^{d-1}$, and where the operators are integral operators that we approximate by the Nyström method. So suppose $d\ge 2$, let $X \subset \mathbb {R}^{d-1}$ be a compact set of positive ($(d-1)$-dimensional) Lebesgue measure, and $K\in \mathcal {L}(C(X))$ an integral operator with continuous kernel $K(\cdot ,\cdot )$, so that K is compact. Thus, for $\phi \in C(X)$ and $x\in X$,

$$\begin{aligned}{} & {} K\phi (x) = \int _X K(x,y)\phi (y)\, dy = J(K(x,\cdot )\phi ), \quad \text{ where }\nonumber \\{} & {} J(\psi ):= \int _X \psi (y) \, dy, \quad \text{ for } \psi \in C(X). \end{aligned}$$

(3.10)

In the Nyström method we approximate K by replacing the integration functional $J:C(X)\rightarrow \mathbb {C}$ by a sequence of numerical quadrature rules. For each $N \in \mathbb {N}$ we choose points $x_{q,N}\in X$ and weights¹²$\omega _{q,N}\ge 0$, for $q=1,\ldots , N$, and define $J_N:C(X)\rightarrow \mathbb {C}$ by

$$\begin{aligned} J_N(\psi ) := \sum _{q=1}^N \omega _{q,N}\psi (x_{q,N}), \qquad \psi \in C(X), \end{aligned}$$

(3.11)

and a Nyström approximation $K_N\in \mathcal {L}(C(X))$ to K by

$$\begin{aligned} K_N \phi (x) := J_N(K(x,\cdot )\phi )) = \sum _{q=1}^N \omega _{q,N}K(x,x_{q,N}) \phi (x_{q,N}), \quad x \in X, \;\; \phi \in C(X). \end{aligned}$$

(3.12)

We will assume that the sequence of quadrature rules is convergent, by which we mean that

$$\begin{aligned} J_N\rightarrow J, \quad \text{ i.e. } \quad J_N\psi \rightarrow J\psi , \quad \text{ for } \text{ all } \psi \in C(X). \end{aligned}$$

(3.13)

This implies (e.g., [2, Proposition 2.1, 2.2]) that $K_N \rightarrow K$ and that the sequence $(K_N)_{N\in \mathbb {N}}\subset \mathcal {L}(C(X))$ is collectively compact, so that (3.9) holds and Theorem 3.6 is applicable. We will also assume that, for each N, $J_N \phi = J \phi$ if $\phi \in C(X)$ is constant, i.e. that

$$\begin{aligned} \sum _{q=1}^N \omega _{q,N} = |X|, \end{aligned}$$

(3.14)

where |X| denotes the Lebesgue measure of X.

Define $\widehat{K}_N:\mathbb {C}^N\rightarrow C(X)$ by

$$\begin{aligned} \widehat{K}_N v(x):= \sum _{q=1}^N \omega _{q,N}K(x,x_{q,N})v_q, \end{aligned}$$

for $x \in X$ and $v=(v_1,...,v_N)\in \mathbb {C}^N$. Moreover, let $P_N :C(X)\rightarrow \mathbb {C}^N$ be defined by $P_N\phi (q):= \phi (x_{q,N})$, for $q=1,...,N$, $\phi \in C(X)$, and define $A_N :\mathbb {C}^N \rightarrow \mathbb {C}^N$ by $A_N:= P_N\hat{K}_N$, so that the matrix entries of $A_N$ are given by

$$\begin{aligned} A_N(p,q) = \omega _{q,N}K(x_{p,N},x_{q,N}), \quad p,q=1,\ldots ,N. \end{aligned}$$

(3.15)

In the following we will use $\Vert \cdot \Vert _\infty$ to denote all of: i) the standard supremum norm on C(X); ii) the standard infinity norm on $\mathbb {C}^N$; iii) the induced operator norm of an operator on C(X); iv) the infinity norm of a square matrix.

Lemma 3.7

The following inequalities and equalities hold for all $N \in \mathbb {N}$:

• (i)
  $$\begin{aligned}{} & {} \max _{1\le p\le N} \sum _{q=1}^N \omega _{q,N}|K(x_{p,N},x_{q,N})| = \left\Vert A_N\right\Vert _\infty \le \left\Vert K_N\right\Vert _\infty \\ {}{} & {} = \max _{x\in X} \sum _{q=1}^N \omega _{q,N}|K(x,x_{q,N})|; \end{aligned}$$

• (ii) $\sigma (K_N) = \{0\} \cup \sigma (A_N)$;

• (iii) for $\lambda \in \mathbb {C}\setminus \sigma (K_N)$,

  $$\begin{aligned} \max \left( |\lambda |^{-1},\left\Vert (A_N - \lambda I)^{-1}\right\Vert _\infty \right)&\le \left\Vert (K_N - \lambda I)^{-1}\right\Vert _\infty \\&\le |\lambda |^{-1}\left( 1+\left\Vert K_N\right\Vert _\infty \left\Vert (A_N - \lambda I)^{-1}\right\Vert _\infty \right) . \end{aligned}$$

Proof

The first equality in (i) is the standard explicit formula for the infinity norm of a matrix. The last equality is proved similarly, and then (i) is clear (or see [23, Lemma 4.7.17]). That the spectra of $K_N$ and $A_N$ coincide on $\mathbb {C}\setminus \{0\}$ is standard (e.g., [23, Lemma 4.7.18]), and $0\in \sigma (K_N)$ since $K_N$ is compact and C(X) is infinite-dimensional, so (ii) holds. Part (iii) is a combination of Hackbusch [23, Lemma 4.7.18] and Anselone [2, Proposition 2.3], plus the facts that $0\in \sigma (K_N)$ and (e.g., [14, Th. 1.2.10]) $\Vert (K_N - \lambda I)^{-1}\Vert \ge (\mathop {\textrm{dist}}\limits (\lambda ,\sigma (K_N)))^{-1}$. $\square$

Note that, by Lemma 3.7(i) and (3.14),

$$\begin{aligned} \Vert A_N\Vert _\infty \le \Vert K_N\Vert _\infty \le K_{\textrm{max}} |X|, \quad \text{ where } \quad K_{\textrm{max}}:= \max _{x,y\in X}|K(x,y)|. \end{aligned}$$

(3.16)

Commonly, for computational efficiency or otherwise, and this is the case in Sects. 4.2 and 4.3, we approximate $K(\cdot ,\cdot )$ by another continuous kernel $K^{\dag }(\cdot ,\cdot )$ in (3.12) and (3.15). Let $K_N^{\dag }\in \mathcal {L}(C(X))$ denote the operator defined by the right hand side of (3.12) with K replaced by $K^{\dag }$, and similarly $A_N^{\dag }$ denote the matrix (3.15) with K replaced by $K^{\dag }$. Lemma 3.7(i) and (3.14) imply that

$$\begin{aligned} \Vert A_N-A_N^{\dag }\Vert _\infty \le \Vert K_N-K_N^{\dag }\Vert _\infty \le e^{\dag } |X|, \quad \text{ where } \quad e^{\dag }:= \max _{x,y\in X}|K(x,y)-K^{\dag }(x,y)|. \end{aligned}$$

(3.17)

The following theorem (cf. Corollary 3.5) follows in large part from Lemmas 3.2, 3.4, and 3.7.

Theorem 3.8

Suppose that $\rho _0 > 0$, $N\in \mathbb {N}$, and $\rho (A_N^{\dag })< \rho _0$, and recursively define $\mu _\ell$, for $\ell \in \mathbb {N}$, by $\mu _{1}:= \rho _0$, and by

$$\begin{aligned} \nu _{\ell } := \left\Vert (A_{N}^{\dag } - \mu _{\ell } I)^{-1}\right\Vert _{\infty }^{-1} \quad \textrm{and} \quad \mu _{\ell +1} := \mu _{\ell }e^{i\frac{\nu _{\ell }}{2\rho _0}}, \quad \textrm{for } \,\ell \in \mathbb {N}. \end{aligned}$$

(3.18)

Further, let $n_N \in \mathbb {N}$ be the smallest integer such that $\sum _{\ell = 1}^{n_N} \nu _{\ell }\ge 4\pi \rho _0$, and let

$$\begin{aligned} R_N := \min _{\ell =1,\ldots ,n_N} \left( \frac{\nu _\ell }{4}+\frac{\nu _{\ell +1}}{2}\right) . \end{aligned}$$

(3.19)

If

$$\begin{aligned} \left\Vert (K-K_N)K\right\Vert _\infty < \rho _0\left( \rho _0\left( 1+\Vert K^{\dag }_N\Vert _\infty R_N^{-1}\right) ^{-1} - \Vert K_N-K_N^{\dag }\Vert _\infty \right) , \end{aligned}$$

(3.20)

or $\Vert A_N-A_N^{\dag }\Vert _\infty < R_N$ and

$$\begin{aligned} \left\Vert (K-K_N)K\right\Vert _\infty < \rho _0^2\left( 1+\Vert K_N\Vert _\infty (R_N-\Vert A_N-A_N^{\dag }\Vert _\infty )^{-1}\right) ^{-1}, \end{aligned}$$

(3.21)

then $\rho (K)<\rho _0$. Conversely, if $\rho (K)<\rho _0$, provided $e^{\dag }$ defined by (3.17) is sufficiently small, (3.20), (3.21), and $\rho (A_N^{\dag })<\rho _0$ hold for all sufficiently large N.

Proof

If $\rho _0>0$ and $\rho (A_N^{\dag })< \rho _0$ then, by Lemma 3.4, $\Vert (A_N^{\dag }-\lambda I)^{-1}\Vert _\infty \le R_N^{-1}$ for $\lambda \in \rho _0\mathbb {T}$, so that $\Vert (K_N^{\dag }-\lambda I)^{-1}\Vert _\infty \le r^{-1}_0(1+\Vert K_N^{\dag }\Vert _\infty R_N^{-1})$, by Lemma 3.7(iii). Thus $\rho (K)<\rho _0$ if (3.20) holds, by Lemma 3.2 applied with $Y=C(X)$, $T=K$, $S=K_N$, $\widehat{S} = K_N^{\dag }$, and $F=\rho _0 \mathbb {T}$. Further, for $\lambda \in \rho _0 \mathbb {T}$, the first of the above bounds and (3.1) implies that $R_N-\Vert A_N-A_N^{\dag }\Vert _\infty \le \Vert (A_N-\lambda I)^{-1}\Vert ^{-1}_\infty$, so that, by Lemma 3.7(iii), $\Vert (K_N-\lambda I)^{-1}\Vert _\infty \le r^{-1}_0(1+\Vert K_N\Vert _\infty (R_N-\Vert A_N-A_N^{\dag }\Vert _\infty )^{-1})$. Thus $\rho (K)<\rho _0$ if (3.21) holds, by Lemma 3.2 applied with $Y=C(X)$, $T=K$, $\widehat{S} =S=K_N$, and $F=\rho _0 \mathbb {T}$.

To see the converse, note that, by Theorem 3.6, . Indeed [2, Theorem 4.7], there exists $N_0\in \mathbb {N}$ such that $\Vert (K_N-\lambda I)^{-1}\Vert _\infty$ is bounded uniformly in $\lambda$ and N for $N\ge N_0$ and $|\lambda |\ge \rho _0$, which implies, by (3.1) and (3.17), that the same holds for $\Vert (K^{\dag }_N-\lambda I)^{-1}\Vert _\infty$ if $e^{\dag }$ is sufficiently small. This implies, by Lemma 3.7(iii), that the same holds for $\Vert (A^{\dag }_N-\lambda I)^{-1}\Vert _\infty$. Noting Lemma 3.7(ii), it follows that there exists $c>0$ such that, for all sufficiently large N, $\rho (A_N^{\dag })=\rho (K_N^{\dag })< \rho _0$ and $R_N\ge c$. Thus, and applying (3.16) and (3.17), we see that, for some $c^*>0$, the right hand sides of (3.20) and (3.21) are $\ge c^*$ for all sufficiently large N if $e^{\dag }$ is sufficiently small. But also $\Vert (K-K_N)K\Vert \rightarrow 0$ by (3.9), so that (3.20) holds for all sufficiently large N. $\square$

Remark 3.9

(Computational cost as N increases) The argument in the above proof makes clear that, if $\rho (K) < \rho _0$ and $e^{\dag }$ is sufficiently small, then, for some $N_0\in \mathbb {N}$, $\Vert (A_N^{\dag }-\lambda I)^{-1}\Vert _\infty$ is bounded uniformly in $\lambda$ and N for $\lambda \in \rho _0 \mathbb {T}$ and $N\ge N_0$. This in turn implies that, for some $n_{\max }\in \mathbb {N}$, $n_N\le n_{\max }$ for $N\ge N_0$. Thus the computational cost of evaluation of $R_N$ given by (3.19) is $O(N^3)$, the cost of inverting $n_N\le n_{\max }$ order N matrices by classical direct methods.

Remark 3.10

(Comparison of (3.20) and (3.21)) Let $\textrm{RHS}_1$ and $\textrm{RHS}_2$ denote the right hand sides of (3.20) and (3.21), respectively. If $K_N^{\dag }=K_N$ (so $A_N^{\dag }=A_N$), then $\textrm{RHS}_2=\textrm{RHS}_1$. If $K^{\dag }_N\ne K_N$ with $\Vert A_N-A_N^{\dag }\Vert _\infty <R_N$, then, where $\mathcal {D}:= \rho _0^{-2}(R_N+\Vert K_N\Vert _\infty - \Vert A_N-A_N^\dag \Vert _\infty )(\textrm{RHS}_2-\textrm{RHS}_1)$,

$$\begin{aligned} \mathcal {D}= & {} \rho _0^{-1}\Vert K_N-K_N^{\dag }\Vert _\infty (\Vert K_N\Vert _\infty -\rho _0 + R_N-\Vert A_N-A_N^\dag \Vert _\infty ) \, \\{} & {} +\,\frac{\Vert K_N^\dag \Vert _\infty (\Vert K_N-K_N^\dag \Vert _\infty -\Vert A_N-A_N^\dag \Vert _\infty )+R_N(\Vert K_N-K_N^\dag \Vert _\infty + \Vert K_N^\dag \Vert _\infty -\Vert K_N\Vert _\infty )}{R_N+\Vert K_N^\dag \Vert _\infty }, \end{aligned}$$

so that $\textrm{RHS}_2-\textrm{RHS}_1> \rho _0\Vert K_N-K_N^{\dag }\Vert _\infty (\Vert K_N\Vert _\infty -\rho _0)/(R_N+\Vert K_N\Vert _\infty - \Vert A_N-A_N^{\dag }\Vert _\infty )$, recalling (3.17). Thus (3.20) implies (3.21) if $\Vert A_N-A_N^{\dag }\Vert _\infty <R_N$ and $\rho _0\le \Vert K_N\Vert _\infty$. Note that [2, Theorem 2.13] $\Vert K_N\Vert _\infty \rightarrow \Vert K\Vert$ as $N\rightarrow \infty$ and it is $\rho _0<\Vert K\Vert _\infty$ for which Theorem 3.8 is arguably of most interest, as $\rho (K) \le \Vert K\Vert _\infty$, and we may be able to estimate $\Vert K\Vert _\infty$ sharply by other methods.

Remark 3.11

(The matrix case) In Sect. 4.3 we will apply the above results, in particular Theorem 3.8, in a case where K is a $2\times 2$ matrix of integral operators on C(X) with continuous kernels, and $K_N$ is its Nyström method approximation defined by approximating each integral operator in the $2\times 2$ matrix as in (3.12). The matrix $A_N$ is then a $2N\times 2N$ matrix consisting of four $N\times N$ blocks each defined as in (3.15). Parts (ii) and (iii) of Lemma 3.7 apply in this case, as does (i) in a straightforwardly modified form, in particular we still have that $\Vert A_N\Vert _\infty \le \Vert K_N\Vert _\infty$. (Here $\Vert A_N\Vert _\infty$ is the usual infinity norm of the matrix $A_N$ and $\Vert K_N\Vert _\infty$ is the norm of $K_N$ as an operator on $(C(X))^2$, which we equip with the norm defined by $\Vert (\phi _1,\phi _2)\Vert _\infty := \max \{\Vert \phi _1\Vert _\infty ,\Vert \phi _2\Vert _\infty \}$, for $(\phi _1,\phi _2)\in C(X)^2$). Thus Theorem 3.8, which depends on Lemma 3.7(ii) and (iii) and the general Banach space results of Sect. 3.1, still applies. One way to see the validity of Lemma 3.7(ii) and (iii) in this matrix case is to argue as follows. Choose $x^*\in \mathbb {R}^{d-1}$ such that $X^\prime := X+x^*$ does not intersect X. K is equivalent, through an obvious isometric isomorphism, to a matrix operator $K^\prime$ on $C(X)\times C(X^\prime )$, which is in turn equivalent, through another obvious isometric isomorphism, to a single integral operator $\widetilde{K}$ on $C(X\cup X^\prime )$. Lemma 3.7 applies to $\widetilde{K}$ and its Nyström method approximation, so that (ii) and (iii) of this lemma [and (i) in modified form] apply to K.

4 The double-layer operator on dilation invariant graphs

Let $A \subseteq \mathbb {R}^{d-1}$ be an open cone, $f :A \rightarrow \mathbb {R}$ a Lipschitz continuous function, and consider the graph $\Gamma = \left\{ (x,f(x)): x \in A\right\} \subset \mathbb {R}^d$ in the case that $\alpha \Gamma = \Gamma$, for some $\alpha \in (0,1)$; that is, in the case that

$$\begin{aligned} f(\alpha x) = \alpha f(x), \quad x\in A. \end{aligned}$$

(4.1)

We will term such graphs dilation invariant.

In this section, the largest of the paper, we study the spectrum and essential spectrum of the double-layer (DL) operator $D_{\Gamma } :L^2(\Gamma ) \rightarrow L^2(\Gamma )$ given by (1.2) on such graphs. (The case $A=\mathbb {R}^{d-1}$ is of particular interest for later applications.) In Sect. 4.1 we show, for general dimension $d\ge 2$, that (as operators on $L^2(\Gamma )$) $\sigma (D_\Gamma )=\sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma )$ (and, similarly, that $W(D_\Gamma )=W_{{{\,\textrm{ess}\,}}}(D_\Gamma )$) and, by Floquet–Bloch-transform arguments, that $\sigma (D_\Gamma )$ is the union of the spectra of a family of operators $K_t:L^2(\Gamma _0)\rightarrow L^2(\Gamma _0)$, for $t\in [-\pi ,\pi ]$, where $\Gamma _0$ is a particular relatively closed and bounded subset of $\Gamma$. Moreover, helpful for the later computation of $\sigma (K_t)$, each $K_t$ is compact, and so has a discrete spectrum, if $f\in C^1(A\setminus \{0\})$ (Corollary 4.6).

In the remaining subsections, Sects. 4.2–4.4, we focus on the 2D case, considering the Nyström approximation of spectral properties of $K_t$, combining the general results of Sect. 3.2 with explicit estimates for the particular operators $K_t$. The case $A=\mathbb {R}$, with f real-analytic on $\mathbb {R}\setminus \{0\}$, is treated in Sect. 4.3. It is this case that is relevant, via, e.g., (1.12), to the computation of $\sigma (D_\Gamma )$ and the spectral radius conjecture when $\Gamma$ is the boundary of a bounded Lipschitz domain. But this case is rather complex; the operator $K_t$ is studied by reducing it to a $2\times 2$ operator matrix, corresponding to the split of $\mathbb {R}\setminus \{0\}$ into the two half-axes $(-\infty ,0)$ and $(0,\infty )$. To get the main ideas across, and prove many of the results we need in a simpler setting, we first study, in Sect. 4.2, the easier case $A=(0,\infty )$ with f real-analytic. Sections 4.2 and 4.3 are concerned with computation of the spectrum and spectral radius of $D_\Gamma$. In Sect. 4.4, related to the question at the end of Sect. 1.1, we also compute lower bounds for $W_{{{\,\textrm{ess}\,}}}(D_\Gamma )$, under the same assumptions on $\Gamma$ as in Sects. 4.2 and 4.3.

4.1 Floquet–Bloch transform results

Let $V_{\alpha } :L^2(\Gamma ) \rightarrow L^2(\Gamma )$ be dilation by $\alpha$, that is,

$$\begin{aligned} V_{\alpha }\phi (x) = \alpha ^{(d-1)/2}\phi (\alpha x), \quad x \in \Gamma . \end{aligned}$$

(4.2)

$V_{\alpha }$ is unitary and commutes with $D_{\Gamma }$: noting that $n(y)=n(\alpha y)$, for $y\in \Gamma$, we see that, for all $\phi \in L^2(\Gamma )$ and almost every $x \in \Gamma$,

$$\begin{aligned} D_{\Gamma }V_{\alpha }\phi (x)&= \frac{1}{c_d}\int _{\Gamma } \frac{(x-y) \cdot n(y)}{|x-y|^d} \alpha ^{(d-1)/2}\phi (\alpha y) \, \textrm{d}s(y)\\&= \frac{\alpha ^{(d-1)/2}}{c_d}\int _{\Gamma } \frac{(x-\alpha ^{-1}y) \cdot n(\alpha ^{-1}y)}{|x-\alpha ^{-1}y|^d} \phi (y) \, \alpha ^{-(d-1)}\textrm{d}s(y)\\&= \frac{\alpha ^{(d-1)/2}}{c_d}\int _{\Gamma } \frac{(\alpha x-y) \cdot n(y)}{|\alpha x-y|^d} \phi (y) \, \textrm{d}s(y)\\&= V_{\alpha }D_{\Gamma }\phi (x). \end{aligned}$$

This already implies that the spectrum and the essential spectrum of $D_{\Gamma }$ coincide, as implied by the following simple proposition (cf. [11, Lemma 2.7]).

Proposition 4.1

Let H be a Hilbert space and $T\in \mathcal {L}(H)$. If T commutes with a sequence of unitary operators $(U_j)_{j \in \mathbb {N}}$ that converges weakly to 0, then $\sigma (T) = \sigma _{{{\,\textrm{ess}\,}}}(T)$.

Proof

Assume there exists $\phi \in H \setminus \left\{ 0\right\}$ such that $T\phi = 0$. As T and $U_j$ commute, also $TU_j\phi = 0$ for all $j \in \mathbb {N}$. In particular, $\left\{ U_j\phi : j \in \mathbb {N}\right\} \subseteq \ker (T)$. As the operators $U_j$ are unitary and $U_j \rightarrow 0$ weakly, the sequence $(U_j\phi )_{j \in \mathbb {N}}$ cannot have a convergent subsequence. Hence, $\ker (T)$ is either trivial or infinite-dimensional. Similarly, $\ker (T')$ is either trivial or infinite-dimensional. Thus, if T is Fredholm, it is invertible. Considering $T - \lambda I$ instead of T yields the result. $\square$

Corollary 4.2

Let $\Gamma \subset \mathbb {R}^d$ be a dilation invariant graph. Then $\sigma (D_{\Gamma }) = \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma })$, $\overline{W(D_{\Gamma })} = W_{{{\,\textrm{ess}\,}}}(D_{\Gamma })$, and $\left\Vert D_{\Gamma }\right\Vert _{{{\,\textrm{ess}\,}}} = \left\Vert D_{\Gamma }\right\Vert$.

Proof

Assume that $\phi ,\psi \in L^2(\Gamma )$ have compact support and $0 \notin {{\,\textrm{supp}\,}}\phi \cup {{\,\textrm{supp}\,}}\psi$. Then

$$\begin{aligned} \left\langle V_{\alpha }^j\phi , \psi \right\rangle = \int _{\Gamma } \alpha ^{j(d-1)/2}\phi (\alpha ^j x) \overline{\psi (x)} \, \textrm{d}s(x). \end{aligned}$$

If $\left|j\right|$ is sufficiently large, the integrand vanishes. Because compactly supported functions are dense in $L^2(\Gamma )$, it follows that $V_{\alpha }^j\rightarrow 0$ weakly as $\left|j\right| \rightarrow \infty$. The equality of spectrum and essential spectrum follows from Proposition 4.1. The results for the numerical range and norm follow from Chandler-Wilde and Spence [11, Lemma 2.7]. $\square$

To make use of standard Floquet–Bloch/Fourier transform results, it is convenient to view $D_\Gamma$ as a discrete convolution operator. For $j\in \mathbb {Z}$ let $\Gamma _j:= \left\{ ({\tilde{x}},f({\tilde{x}})) \in \Gamma : |{\tilde{x}}| \in [\alpha ^{j+1},\alpha ^j]\right\}$. We can identify $L^2(\Gamma _j)$ with a closed subspace of $L^2(\Gamma )$ by extending by 0. Let $P_j :L^2(\Gamma ) \rightarrow L^2(\Gamma _j)$ denote orthogonal projection. Clearly,

$$\begin{aligned} P_j\phi (x) = {\left\{ \begin{array}{ll} \phi (x) &{} \text {if } x \in \Gamma _j, \\ 0 &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

and we note that $P_jV_{\alpha } = V_{\alpha }P_{j+1}$, so that also $P_jV_\alpha ^k = V_\alpha ^k P_{j+k}$, $k\in \mathbb {Z}$. Let

$$\begin{aligned} D_j:= V_{\alpha }^jP_jD_{\Gamma }|_{L^2(\Gamma _0)}: L^2(\Gamma _0) \rightarrow L^2(\Gamma _0), \end{aligned}$$

so that

$$\begin{aligned} D_j\phi (x) = \alpha ^{j(d-1)/2} D_{\Gamma }\phi (\alpha ^jx), \quad x\in \Gamma _0, \quad \phi \in L^2(\Gamma _0). \end{aligned}$$

(4.3)

Let $\mathcal {G}:L^2(\Gamma )\rightarrow \ell ^2(\mathbb {Z}, L^2(\Gamma _0))$ be the unitary operator.¹³$\phi \mapsto (V_\alpha ^jP_j\phi )_{j\in \mathbb {Z}}$, and let $\widetilde{D}_\Gamma := \mathcal {G}D_\Gamma \mathcal {G}^{-1}$. It is a straightforward calculation to see that the action of $\widetilde{D}_\Gamma$ is that of a discrete convolution: for $\psi =(\psi _n)_{n\in \mathbb {Z}}\in \ell ^2(\mathbb {Z}, L^2(\Gamma _0))$, $\widetilde{D}_\Gamma \psi = ((\widetilde{D}_\Gamma \psi )_m)_{m\in \mathbb {Z}}$ where

$$\begin{aligned} (\widetilde{D}_\Gamma \psi )_m = \sum _{n\in \mathbb {Z}} D_{m-n} \psi _n, \quad m\in \mathbb {Z}. \end{aligned}$$

(4.4)

The series in the above definition converges absolutely; indeed $\widetilde{D}_\Gamma$ is an operator in the so-called Wiener algebra, i.e. $\sum _{j\in \mathbb {Z}} \Vert D_j\Vert <\infty$ (e.g., [42, Defintion 1.43]), by the following estimate.

Proposition 4.3

The operators $D_j$ are Hilbert-Schmidt for $|j|\ge 2$ and satisfy the Hilbert-Schmidt norm estimate

$$\begin{aligned} \left\Vert D_j\right\Vert _{\textrm{HS}} \le \frac{1}{c_d} \frac{|\Gamma _0|}{(\alpha - \alpha ^{|j|})^{d-1}}\,\alpha ^{|j|(d-1)/2}, \quad |j| \ge 2, \end{aligned}$$

(4.5)

where $|\Gamma _0|$ denotes the surface measure of $\Gamma _0$.

Proof

For $j \ge 2$ we have

$$\begin{aligned} \int _{\Gamma _j} \int _{\Gamma _0} \left|\frac{(x-y) \cdot n(y)}{|x-y|^d}\right|^2 \, \textrm{d}s(y) \, \textrm{d}s(x) \le \int _{\Gamma _j} \int _{\Gamma _0} \frac{1}{|x-y|^{2d-2}} \, \textrm{d}s(y) \, \textrm{d}s(x) \end{aligned}$$

and $|x-y| \ge \alpha - \alpha ^j$. Hence

$$\begin{aligned} \int _{\Gamma _j} \int _{\Gamma _0} \left|\frac{(x-y) \cdot n(y)}{|x-y|^d}\right|^2 \, \textrm{d}s(y) \, \textrm{d}s(x) \le \frac{|\Gamma _j||\Gamma _0|}{(\alpha - \alpha ^j)^{2d-2}} = \frac{|\Gamma _0|^2\alpha ^{j(d-1)}}{(\alpha - \alpha ^j)^{2d-2}}. \end{aligned}$$

Similarly, for $j \le -2$,

$$\begin{aligned} \int _{\Gamma _j} \int _{\Gamma _0} \left|\frac{(x-y) \cdot n(y)}{|x-y|^d}\right|^2 \, \textrm{d}s(y) \, \textrm{d}s(x) \le \frac{|\Gamma _0|^2\alpha ^{j(d-1)}}{(\alpha ^{j+1} - 1)^{2d-2}} = \frac{|\Gamma _0|^2\alpha ^{-j(d-1)}}{(\alpha - \alpha ^{-j})^{2d-2}}. \end{aligned}$$

Thus, recalling the standard characterisation of the Hilbert-Schmidt norm of integral operators (e.g., [32, Ex. 11.11]), $P_jD_{\Gamma }|_{L^2(\Gamma _0)}$ is Hilbert-Schmidt, with Hilbert-Schmidt norm bounded by the right hand side of (4.5). As unitary operators preserve Hilbert-Schmidt norms, it follows that $D_j$ is Hilbert-Schmidt and that (4.5) holds. $\square$

Given a Hilbert space H let $\mathcal {F}:\ell ^2(\mathbb {Z},H)\rightarrow L^2([-\pi ,\pi ],H)$ denote the operator, often termed in the general Hilbert space case, e.g., [5], a Floquet–Bloch transform, that constructs a Fourier series from coefficients, given by

$$\begin{aligned} (\mathcal {F}\psi )(t) = (2\pi )^{-1/2} \sum _{j\in \mathbb {Z}} e^{i jt} \psi _n, \quad t\in [-\pi ,\pi ], \end{aligned}$$

for $\psi =(\psi _j)_{j\in \mathbb {Z}}\in \ell ^2(\mathbb {Z},H)$. It is standard that this is a unitary operator (e.g., [14, Proof of Theorem 4.4.9]) that diagonalises discrete convolutions (see, e.g., [14, Theorem 4.49] for the case when H is finite-dimensional, [54, Theorem 2.3.25] for the general case). In the case $H=L^2(\Gamma _0)$, defining $\widehat{D}:L^2([-\pi ,\pi ],H)\rightarrow L^2([-\pi ,\pi ],H)$ by $\widehat{D}:= \mathcal {F}\widetilde{D}_\Gamma \mathcal {F}^{-1}$, straightforward calculations yield that

$$\begin{aligned} \langle \widehat{D} \phi ,\psi \rangle _{L^2([-\pi ,\pi ],L^2(\Gamma _0))} = \int _{-\pi }^\pi \langle K_t\phi (t),\psi (t)\rangle \, \textrm{d}t, \quad \phi ,\psi \in L^2([-\pi ,\pi ],L^2(\Gamma _0)), \end{aligned}$$

(4.6)

where

$$\begin{aligned} K_t := \sum \limits _{j = -\infty }^{\infty } e^{ijt}D_j, \quad t\in \mathbb {R}. \end{aligned}$$

(4.7)

(The bounds of Proposition 4.10 imply that $K_t$ is well-defined by (4.7) and depends continuously on t; indeed the mapping $t\mapsto K_t$ is $C^\infty$.) The following characterisation follows immediately from (4.6), the continuity of $t\mapsto K_t$, and Corollary 4.2; note that $\mathop {\textrm{conv}}\limits$ denotes the closed convex hull.

Theorem 4.4

We have

$$\begin{aligned} \Vert D_\Gamma \Vert _{{{\,\textrm{ess}\,}}}&= \Vert D_{\Gamma }\Vert = \max \limits _{t \in [-\pi ,\pi ]} \left\Vert K_t\right\Vert ,\\ W_{{{\,\textrm{ess}\,}}}(D_\Gamma )&= \overline{W(D_{\Gamma })} = \mathop {\textrm{conv}}\limits \left( \bigcup \limits _{t \in [-\pi ,\pi ]} W(K_t)\right) ,\\ \sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma )&=\sigma (D_{\Gamma }) = \bigcup \limits _{t \in [-\pi ,\pi ]} \sigma (K_t). \end{aligned}$$

Proof

That the essential spectrum, numerical range, and norm coincide with their non-essential counterparts is Corollary 4.2. Since $D_\Gamma$ and $\widehat{D}$ are unitarily equivalent, they have the same spectrum, numerical range, and norm. The result thus follows from (4.6); see [54, Theorem 2.3.25] for the case of the spectrum; the argument for the norm and numerical range are similar. $\square$

Remark 4.5

(Symmetry of $K_t$) Where $K_t(\cdot ,\cdot )$ denotes the kernel of $K_t$, $K_{-t}(\cdot ,\cdot )=\overline{K_t(\cdot ,\cdot )}$, for $t\in [-\pi ,\pi ]$. Thus $\Vert K_{-t}\Vert =\Vert K_{t}\Vert$, $W(K_{-t}) = \{{\bar{z}}:z\in W(K_t)\}$, and $\sigma (K_{-t})=\{{\bar{z}}:z\in \sigma (K_t)\}$, for $t\in [-\pi ,\pi ]$, so that, where $w(K_t):= \sup _{z\in W(K_t)}|z|$ is the numerical radius of $K_t$,

$$\begin{aligned} \Vert D_\Gamma \Vert _{{{\,\textrm{ess}\,}}} = \max _{t\in [0,\pi ]}\Vert K_t\Vert ,\quad w_{{{\,\textrm{ess}\,}}}(D_\Gamma ) = \max _{t\in [0,\pi ]} w(K_t), \quad \rho _{{{\,\textrm{ess}\,}}}(D_\Gamma ) = \max _{t\in [0,\pi ]} \rho (K_t). \end{aligned}$$

Our focus in the next subsections will be 2D cases where f is analytic on $A\setminus \{0\}$. In such cases, indeed whenever $\Gamma$ is locally $C^1$ away from 0, $K_t$ is compact for all t as a consequence of standard results on the double-layer operator on $C^1$ domains [19] and Proposition 4.10.

Corollary 4.6

Suppose that $f\in C^1(A\setminus \{0\})$ so that $\Gamma$ is locally $C^1$ at each $x\in \Gamma \setminus \{0\}$. Then $K_t$ is compact for every $t \in \mathbb {R}$.

Proof

Let $Q = P_{-1} + P_0 + P_1$. Then $QD_{\Gamma }|_{L^2(\Gamma _{-1} \cup \Gamma _0 \cup \Gamma _1)}$ is the DL operator on $\Gamma _{-1} \cup \Gamma _0 \cup \Gamma _1$. It thus follows from [19, Theorem 1.2] that $QD_{\Gamma }|_{L^2(\Gamma _{-1} \cup \Gamma _0 \cup \Gamma _1)}$ is compact. As, for $j = -1, 0, 1$,

$$\begin{aligned} D_j = V_{\alpha }^jP_jD_\Gamma |_{L^2(\Gamma _0)} = V_{\alpha }^jP_jQD_\Gamma |_{L^2(\Gamma _0)}, \end{aligned}$$

$D_j$ is compact for $j = -1,0,1$. The compactness of $K_t$ thus follows from Proposition 4.3. $\square$

4.2 The 2D case: one-sided infinite graphs

We continue to assume that $\Gamma$ is a dilation invariant graph, as defined at the start of Sect. 4, but specialise now to the case where $d=2$ and the cone is the half-axis $A=\mathbb {R}_+:= (0,\infty )$. Thus $\Gamma = \left\{ (x,f(x)): x \in \mathbb {R}_+\right\}$ and, for some $\alpha \in (0,1)$, $f(\alpha x) = \alpha f(x)$, $x\in \mathbb {R}_+$. Our starting point is Corollary 4.4 which expresses $\sigma (D_\Gamma )$ as the union of the spectra of the operators $K_t$, $t\in [-\pi ,\pi ]$. Recall that these operators are compact if $f\in C^1(\mathbb {R}_+)$. Our goal is to apply the Nyström method and the results of Sect. 3.2 to compute spectral properties of $K_t$ and hence of $D_\Gamma$ in the case that $f\in \mathcal {A}(\mathbb {R}_+)$, the space of functions $\mathbb {R}_+\rightarrow \mathbb {R}$ that are real analytic (a prototypical example is Fig. 2). Our standing assumption through this subsection is that

$$\begin{aligned} \Gamma= & {} \left\{ (x,f(x)): x \in \mathbb {R}_+\right\} \; \text{ where } \; f\in \mathcal {A}(\mathbb {R}_+) \; \text{ and, } \text{ for } \text{ some }\, \nonumber \\{} & {} \quad \alpha \in (0,1), \; f(\alpha x) = \alpha f(x), \quad x\in \mathbb {R}_+. \end{aligned}$$

(4.8)

Fig. 2

Graph of $f :\mathbb {R}_+ \rightarrow \mathbb {R}$, $f(x):= x\sin ^2(\pi \log _{\alpha }(x))$, for $\alpha = \frac{3}{4}$

Notably, via approximations of the spectrum of $K_t$ for each t, we will obtain (see Theorem 4.13) a Nyström approximation $\sigma ^N(D_\Gamma )$, for $\sigma (D_\Gamma )=\sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma )$, which is $\{0\}$ plus the union over finitely many $t\in [-\pi ,\pi ]$ of the spectra of $N\times N$ matrices $A^M_{t,N}$, where each $A^M_{t,N}$ is obtained via Nyström discretisation of a unitary transformation, ${\tilde{K}}_t$, of $K_t$. Our first main result, Theorem 4.13, is to show that as $N\rightarrow \infty$. Our other, more substantial result (Theorem 4.15) is to develop a fully discrete algorithm to test whether the spectral radius conjecture holds for $\Gamma$, i.e. to test whether $\sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma )=\sigma (D_\Gamma )<\frac{1}{2}$. This algorithm, which derives from Theorem 3.8, requires the computation only of the spectral radii of finitely many finite matrices plus the norms of finitely many finite matrix resolvents.

Before we begin our analysis we note the following equivalences to (4.8) that will play a key role in our calculations. Here, and throughout, the notations

$$\begin{aligned} \Sigma _c := \left\{ z \in \mathbb {C}: {{\,\textrm{Im}\,}}z \in (-c,c)\right\} , \quad \Sigma _0 := \mathbb {R}, \end{aligned}$$

(4.9)

for $c>0$ will be convenient.

Lemma 4.7

Given $f:\mathbb {R}_+\rightarrow \mathbb {R}$ and $\alpha \in (0,1)$, define $g:\mathbb {R}\rightarrow \mathbb {R}$ by

$$\begin{aligned} g(x) := \alpha ^{-x}f(\alpha ^x), \quad x\in \mathbb {R}, \quad \text{ so } \text{ that } \quad f(x) = xg(\log _\alpha x), \quad x\in \mathbb {R}_+. \end{aligned}$$

(4.10)

Then the following are equivalent:

• (i) f is real analytic on $\mathbb {R}_+$ and $f(\alpha x) = \alpha f(x)$, $x>0$;

• (ii) $g:\mathbb {R}\rightarrow \mathbb {R}$ is real analytic and $g(x+1)=g(x)$, $x\in \mathbb {R}$;

• (iii) for some $c>0$, g has an analytic extension to $\Sigma _c$ that satisfies $g(z+1)=g(z)$, for $z\in \Sigma _c$, and g and its derivatives $g^\prime$ and $g^{\prime \prime }$ are bounded in $\Sigma _c$.

Note that if f satisfies our standing assumption (4.8), then, defining g by (4.10),

$$\begin{aligned} f^\prime (x) = g(\log _\alpha x) + \frac{g^\prime (\log _\alpha x)}{\log \alpha }, \quad f^{\prime \prime }(x) = \frac{g^\prime (\log _\alpha x)}{x\log \alpha }+\frac{g^{\prime \prime }(\log _\alpha x)}{x\log ^2 \alpha }, \quad x>0. \end{aligned}$$

(4.11)

It follows from (4.10), the first of (4.11), and the equivalence of i) and iii), that f is Lipschitz continuous on $[0,\infty )$ if we set $f(0):=0$.

To make use of the results from Sect. 3.2 it is convenient to make a change of variables so that we work with integral operators on [0, 1] rather than $\Gamma _0$. Introducing the unitary transformation $U :L^2(\Gamma _0) \rightarrow L^2(0,1)$ given by

$$\begin{aligned} U\phi (s) :=\phi (\alpha ^s,f(\alpha ^s))(1+f'(\alpha ^s)^2)^{1/4}\alpha ^{s/2}\left|\log \alpha \right|^{1/2}, \quad s \in [0,1], \quad \phi \in L^2(\Gamma _0), \end{aligned}$$

(4.12)

define

$$\begin{aligned} \tilde{K}_t := UK_tU^{-1} = \sum _{j=-\infty }^\infty e^{ijt} UD_jU^{-1}, \quad t\in \mathbb {R}. \end{aligned}$$

(4.13)

Straightforward computations, starting from (4.13), (4.3), and (1.4), give that

$$\begin{aligned} {\tilde{K}}_t \phi (x)&= \int _0^1 {\tilde{K}}_t(x,y) \phi (y)\, \textrm{d}y, \quad x\in [0,1], \;\; t\in \mathbb {R}, \end{aligned}$$

(4.14)

where, for $t\in \mathbb {R}$, and $x,y\in \mathbb {R}$ with $x-y\not \in \mathbb {Z}$,

$$\begin{aligned} \tilde{K}_t(x,y) = \frac{1}{2\pi }\sum \limits _{j = -\infty }^{\infty } e^{ijt}\,\frac{p_j(x,y)}{1+q_j(x,y)^2}\left( \frac{1+f'(\alpha ^{x})^2}{1+f'(\alpha ^y)^2}\right) ^{1/4}\alpha ^{\frac{x+y+j}{2}} \left|\log \alpha \right|, \end{aligned}$$

(4.15)

with

$$\begin{aligned}{} & {} p_j(x,y) := \frac{(\alpha ^y-\alpha ^{x+j})f^\prime (\alpha ^y) + f(\alpha ^{x+j}) - f(\alpha ^y)}{(\alpha ^{x+j}-\alpha ^y)^2},\nonumber \\ {}{} & {} \quad q_j(x,y) := \frac{f(\alpha ^{x+j})-f(\alpha ^y)}{\alpha ^{x+j}-\alpha ^y}. \end{aligned}$$

(4.16)

By Taylor’s theorem applied to $F(t):= f((1-t)\alpha ^y+t\alpha ^{x+j})$, we see that, for the same range of x and y,

$$\begin{aligned} q_j(x,y)&= \int _0^1 f^\prime ((1-t)\alpha ^y+t\alpha ^{x+j})\, \textrm{d}t, \end{aligned}$$

(4.17)

$$\begin{aligned} p_j(x,y)&= \int _0^1 f^{\prime \prime }((1-t)\alpha ^y+t\alpha ^{x+j})(1-t)\, \textrm{d}t. \end{aligned}$$

(4.18)

Using (4.17) and (4.18) to extend the definitions of $q_j(x,y)$ and $p_j(x,y)$ to $\{(x,y)\in \mathbb {R}^2:x-y\in \mathbb {Z}\}$,¹⁴ we see that, for each $t\in \mathbb {R}$, each term in the sum (4.15) is continuous on $\mathbb {R}^2$. Further, using (4.11) and the equivalence of (i) and (iii) in Lemma 4.7, it is easy to see that $p_j(x,y) = O(1)$ as $j\rightarrow \infty$, $=O(\alpha ^{-j})$ as $j\rightarrow -\infty$, uniformly for x and y in compact subsets of $\mathbb {R}$, so that the series (4.15) converges absolutely and uniformly on compact subsets, so that ${\tilde{K}}_t(\cdot ,\cdot )\in C(\mathbb {R}^2)$. Further, $f(\alpha x) = \alpha f(x)$ implies that $\tilde{K}_t(x+1,y) = e^{-it}\tilde{K}_t(x,y)$ and $\tilde{K}_t(x,y+1) = e^{it}\tilde{K}_t(x,y)$, $x,y \in \mathbb {R}$. Note that ${\tilde{K}}_t(\cdot ,\cdot )\in C(\mathbb {R}^2)$ implies that ${\tilde{K}}_t:L^2(0,1)\rightarrow C[0,1]$, that ${\tilde{K}}_t$ is compact as an operator on both C[0, 1] and $L^2(0,1)$, and that the spectrum of ${\tilde{K}}_t$ is the same on C[0, 1] as on $L^2(0,1)$, so that

$$\begin{aligned} \sigma (K_t;L^2(\Gamma _0)) = \sigma ({\tilde{K}}_t;L^2(0,1))=\sigma ({\tilde{K}}_t;C[0,1]), \quad t\in \mathbb {R}. \end{aligned}$$

(4.19)

We will use the above equivalence, and the Nyström method results from Sect. 3.2, to compute the spectrum and spectral radius of $K_t$, in particular by applying Theorem 3.8. As a step towards estimating the left hand side of (3.20) in the case when $K={\tilde{K}}_t$ and $K_N$ is its Nyström method approximation, we first show that, under our standing assumption (4.8), ${\tilde{K}}_t(x,\cdot )$ and ${\tilde{K}}_t(\cdot ,y)$ can be extended to bounded analytic functions on $\Sigma _c$ for some $c>0$. We will use the notation $\left\Vert \cdot \right\Vert _c$ for the sup norm on $\Sigma _c$ and, for functions $h :\Sigma _c \times \Sigma _d \rightarrow \mathbb {C}$, the notation

$$\begin{aligned} \left\Vert h\right\Vert _{c,d}:= \sup \limits _{\begin{array}{c} x \in \Sigma _c \\ y \in \Sigma _d \end{array}} \left|h(x,y)\right|. \end{aligned}$$

In an extension of these notations, for $h :\Sigma _c \times \Sigma _c \rightarrow \mathbb {C}$ we define

$$\begin{aligned} \left\Vert h\right\Vert _{c,0}:= \sup \limits _{\begin{array}{c} x \in \Sigma _c \\ y \in \mathbb {R} \end{array}} \left|h(x,y)\right| \quad \text {and} \quad \left\Vert h\right\Vert _{0,c}:= \sup \limits _{\begin{array}{c} x \in \mathbb {R}\\ y \in \Sigma _c \end{array}} \left|h(x,y)\right|, \end{aligned}$$

and, for functions $h:\Sigma _c\rightarrow \mathbb {C}$, we set $\Vert h\Vert _0:=\Vert h\Vert _\infty :=\sup _{x\in \mathbb {R}}|h(x)|$, the ordinary sup norm on $\mathbb {R}$.

Let us now estimate the norms $\Vert \tilde{K}_t\Vert _{c,0}$ and $\Vert \tilde{K}_t\Vert _{0,c}$ under our standing assumption (4.8). Since g is real-valued on $\mathbb {R}$, it follows, using Lemma 2.1, that (4.21) holds for all sufficiently small $c>0$.

Proposition 4.8

Given (4.8), define g by (4.10) so that, by Lemma 4.7, g has an analytic continuation to $\Sigma _c$, for some $c>0$, such that g, $g^\prime$, and $g^{\prime \prime }$ are bounded on $\Sigma _c$ and $g(z+1)=g(z)$, $z\in \Sigma _c$. Let $\mathfrak {I}_c:= \Vert {{\,\textrm{Im}\,}}g\Vert _c + \Vert {{\,\textrm{Im}\,}}g^\prime \Vert _c/|\log \alpha |$, and set

$$\begin{aligned} \mathfrak {F}_d := \Vert g\Vert _d + \Vert g^\prime \Vert _d/|\log \alpha | \quad \text{ and } \quad \mathfrak {G}_d := \Vert g^\prime \Vert _d + \Vert g^{\prime \prime }\Vert _d/|\log \alpha |, \quad \text{ for } d=0,c. \end{aligned}$$

(4.20)

If

$$\begin{aligned} c \le \arccos (\alpha )/\left|\log \alpha \right| \quad \text{ and } \quad \mathfrak {I}_c < 1, \end{aligned}$$

(4.21)

then $\tilde{K}_t(x,\cdot )$ and $\tilde{K}_t(\cdot ,y)$ extend to bounded analytic functions on $\Sigma _c$ for all $x,y,t\in \mathbb {R}$, and

$$\begin{aligned} \Vert \tilde{K}_t\Vert _{c,0}&\le \frac{\left( 1+\mathfrak {F}_c^2\right) ^{1/4}}{\pi \left( 1- \mathfrak {I}_c^2\right) } \Bigg [ \mathfrak {G}_c\, \frac{1+\alpha ^{1/2}+\alpha ^{-1/2}}{4\alpha ^2} + |\log \alpha |(\mathfrak {F}_c+\mathfrak {F}_0) \, \sum _{j=2}^\infty \frac{\alpha ^{j/2}}{\alpha -\alpha ^j}\Bigg ], \end{aligned}$$

(4.22)

$$\begin{aligned} \Vert \tilde{K}_t\Vert _{0,c}&\le \frac{\left( 1+\mathfrak {F}_0^2\right) ^{1/4}}{\pi \left( 1- \mathfrak {I}_c^2\right) ^{5/4}} \Bigg [ \mathfrak {G}_c\, \frac{1+\alpha ^{1/2}+\alpha ^{-1/2}}{4\alpha ^2} + 2|\log \alpha |\,\mathfrak {F}_c \, \sum _{j=2}^\infty \frac{\alpha ^{j/2}}{\alpha -\alpha ^j}\Bigg ]. \end{aligned}$$

(4.23)

Moreover, $\mathbb {R}\rightarrow C([0,1]\times [0,1])$, $t \mapsto \tilde{K}_t(\cdot ,\cdot )$ is continuous.

Remark 4.9

For fixed $\alpha \in (0,1)$, let $F(x):= \alpha ^{x/2}/(\alpha -\alpha ^x)$, for $x>0$, so that the jth term in the sum in (4.22) and (4.23) is F(j). Since F is decreasing on $(1,\infty )$ we have that, for $n=2,3,\ldots$ and $0<\alpha <1$,

$$\begin{aligned} \sum _{j=n+1}^\infty \frac{\alpha ^{j/2}}{\alpha -\alpha ^j}\le & {} \mathcal {B}_n(\alpha ) \quad \text{ so } \text{ that } \end{aligned}$$

(4.24)

$$\begin{aligned} \sum _{j=2}^\infty \frac{\alpha ^{j/2}}{\alpha -\alpha ^j}\le & {} \mathcal {B}^*_n(\alpha ):= \sum _{j=2}^n F(j) + \mathcal {B}_n(\alpha ), \end{aligned}$$

(4.25)

where

$$\begin{aligned} \mathcal {B}_{n}(\alpha ):= & {} \int _{n}^\infty F(x) \,\textrm{d}x = \frac{1}{2\alpha ^{1/2}}\int _{n}^\infty \frac{\textrm{d}x}{\sinh \left( |\log \alpha |(x-1)/2\right) }\\ {}= & {} \frac{\log (\tanh ((n-1)|\log \alpha |/4))}{\alpha ^{1/2}\log \alpha }. \end{aligned}$$

We note, for later reference, that

$$\begin{aligned} \mathcal {B}_{n}(\alpha ) \sim 2\alpha ^{n/2-1}/|\log \alpha |, \quad \text{ as } \quad n\rightarrow \infty . \end{aligned}$$

(4.26)

Proof of Proposition 4.8

Using (4.11) it follows that

$$\begin{aligned}{} & {} f(\alpha ^x) = \alpha ^x g(x), \quad f^\prime (\alpha ^x) = g(x) + \frac{g^\prime (x)}{\log \alpha },\nonumber \\ {}{} & {} \quad f^{\prime \prime }(\alpha ^x) = \frac{g^\prime (x)}{\alpha ^x\log \alpha }+\frac{g^{\prime \prime }(x)}{\alpha ^x\log ^2 \alpha }, \quad x\in \mathbb {R}. \end{aligned}$$

(4.27)

Note that the first of the bounds (4.21) implies that $c|\log \alpha | < \pi /2$, so that the assumptions we have made on g mean that (4.10) provides an analytic extension of f from $\mathbb {R}_+$ to the sector of the complex plane

$$\begin{aligned} G_c:=\{re^{i\theta }: r>0, |\theta | <c|\log \alpha |\} = \{\alpha ^z:z\in \Sigma _c\}, \end{aligned}$$

and the Eqs. (4.11) hold for all x in this sector. Further, with this extension, (4.27) holds for all $x\in \Sigma _c$. Since (4.11) holds for all $x\in G_c$, and noting that $G_c$ is convex, we see that the integrals (4.17) and (4.18) are well-defined for all $x,y\in \Sigma _c$, $j\in \mathbb {Z}$, and provide analytic continuations of $p_j$ and $q_j$ to $\Sigma _c\times \Sigma _c$. Further, by the uniqueness of analytic continuation, the equations (4.16) hold for all $x,y\in \Sigma _c$, with $x+j\ne y$.

To complete the proof we will demonstrate that (4.15) provides, for each $x,y\in \mathbb {R}$, analytic continuations of $\tilde{K}_t(\cdot ,y)$ and $\tilde{K}_t(x,\cdot )$ from $\mathbb {R}$ to $\Sigma _c$ which satisfy the bounds (4.22) and (4.23), by showing that each term in (4.15) is well-defined (i.e., that $f^\prime (\alpha ^y)^2\ne -1$, for $y\in \Sigma _c$, and $1+(q_j(x,y))^2\ne 0$ for $x,y\in \Sigma _c$), and that the series (4.15) converges absolutely and uniformly for $(x,y) \in \mathbb {R}\times \Sigma _c$ and $(x,y)\in \Sigma _c \times \mathbb {R}$.

Using (4.27) we see that

$$\begin{aligned} \sup _{z\in G_c}|{{\,\textrm{Im}\,}}f^\prime (z)| = \sup _{y\in \Sigma _c}|{{\,\textrm{Im}\,}}f^\prime (\alpha ^y)| \le \mathfrak {I}_c, \end{aligned}$$

so that

$$\begin{aligned} \inf \limits _{y \in \Sigma _c} \left|1+f'(\alpha ^y)^2\right|&\ge 1 - \sup \limits _{y \in \Sigma _c} \left|{{\,\textrm{Im}\,}}f'(\alpha ^y)\right|^2 \ge 1 - \mathfrak {I}_c^2, \end{aligned}$$

(4.28)

and, using (4.17),

$$\begin{aligned} \inf _{x,y\in \Sigma _c} \left| 1+q_j(x,y)^2\right| \ge 1 - \sup _{z\in G_c}|{{\,\textrm{Im}\,}}f^\prime (z)|^2 \ge 1 - \mathfrak {I}_c^2. \end{aligned}$$

(4.29)

Thus, where $T_j(x,y)$ denotes the j-th term in the sum (4.15), we see that $T_j(x,y)$ is well-defined and analytic for $x,y\in \Sigma _c$, $j\in \mathbb {Z}$. Moreover, we have $T_j(x+1,y)=e^{-it}T_{j+1}(x,y)$ and it follows from $f(\alpha x) = \alpha f(x)$ that $T_j(x,y+1)=e^{it}T_{j-1}(x,y)$, for $x,y\in \Sigma _c$, $j \in \mathbb {Z}$. Thus, to prove absolute and uniform convergence of (4.15) and the bounds (4.22) and (4.23), it suffices to restrict consideration to $x,y \in \Xi _c:= \{z \in \mathbb {C}:$ ${{\,\textrm{Re}\,}}z \in [0,1],$ ${{\,\textrm{Im}\,}}z \in (-c,c)\}$.

We see that $\sup _{x\in \Xi _c}|\alpha ^{x/2}|\le 1$ and, using (4.27), that

$$\begin{aligned} \sup \limits _{x \in [0,1]} \left|1+f'(\alpha ^x)^2\right| \le 1 + \mathfrak {F}_0^2, \quad \sup \limits _{x \in \Xi _c} \left|1+f'(\alpha ^x)^2\right| \le 1 + \mathfrak {F}_c^2. \end{aligned}$$

To obtain a bound on $|T_j(x,y)|$ for $(x,y) \in [0,1] \times \Xi _c$, and for $(x,y) \in \Xi _c \times [0,1]$, it remains to bound $p_j(x,y)$. Let

$$\begin{aligned} G_c^*:= \{re^{i\theta }\!:\! \alpha ^{2}\!\le \! r\!\le \! \alpha ^{-1}, |\theta | <c|\log \alpha |\} = \{\alpha ^z\!:\!{{\,\textrm{Re}\,}}z\in [-1,2], \, {{\,\textrm{Im}\,}}z\!\in \! (-c,c)\}. \end{aligned}$$

It is clear that $(1-t) \alpha ^y+t\alpha ^{x+j}\in G_c^*$ for every $t\in [0,1]$, $|j|\le 1$, $x\in \Xi _c$, and $y\in [0,1]$ if and only if $G_c^*$ is star-shaped with respect to every point in $[\alpha ,1]$, which holds if and only if c satisfies the first of the bounds (4.21). Likewise, $(1-t) \alpha ^y+t\alpha ^{x+j}\in G_c^*$ for every $t\in [0,1]$, $|j|\le 1$, $x\in [0,1]$, and $y\in \Xi _c$ if and only if the first of (4.21) holds. Thus, if (4.21) holds, $|j|\le 1$, and either $(x,y) \in [0,1] \times \Xi _c$ or $(x,y) \in \Xi _c \times [0,1]$, it follows from (4.11) and (4.18) that

$$\begin{aligned} |p_j(x,y)| \le \frac{1}{2} \sup _{z\in G_c^*}|f^{\prime \prime }(z)| \le \frac{\mathfrak {G}_c}{2\alpha ^2|\log \alpha |}. \end{aligned}$$

On the other hand, if $|j|\ge 2$ and $x,y\in \Xi _c$, then it follows from (4.16) (which we have observed above holds for all $x,y\in \Sigma _c$ with $x+j\ne y$) that

$$\begin{aligned} |p_j(x,y)| \le \frac{|f^\prime (\alpha ^y)| + |q_j(x,y)|}{|\alpha ^{x+j}-\alpha ^y|}. \end{aligned}$$

Further, for $j\in \mathbb {Z}$,

$$\begin{aligned} |q_j(x,y)| \le \sup _{z\in G_c} |f^{\prime }(z)| \le \mathfrak {F}_c, \quad x,y\in \Xi _c, \end{aligned}$$

while $|f^\prime (\alpha ^y)| \le \mathfrak {F}_0$, for $y\in [0,1]$, $\le \mathfrak {F}_c$, for $y\in \Xi _c$. Moreover, for $|j|\ge 2$, $x,y\in \Xi _c$,

$$\begin{aligned} |\alpha ^{x-j}-\alpha ^y| \ge \left| \alpha ^{{{\,\textrm{Re}\,}}x - j} - \alpha ^{{{\,\textrm{Re}\,}}y} \right| \ge \left\{ \begin{array}{cc} \alpha -\alpha ^j, &{} j\ge 2, \\ \alpha ^{j+1}-1, &{} j\le -2. \end{array}\right. \end{aligned}$$

Putting these bounds together we see that, for $|j|\le 1$,

$$\begin{aligned} \sup \limits _{x\in \Xi _c, y\in [0,1]}|T_j(x,y)|&\le \frac{\mathfrak {G}_c \left( 1+\mathfrak {F}_c^2\right) ^{1/4}\alpha ^{j/2}}{2\alpha ^2\left( 1- \mathfrak {I}_c^2\right) },\\ \sup \limits _{x\in [0,1], y\in \Xi _c}|T_j(x,y)|&\le \frac{\mathfrak {G}_c \left( 1+\mathfrak {F}_0^2\right) ^{1/4}\alpha ^{j/2}}{2\alpha ^2\left( 1- \mathfrak {I}_c^2 \right) ^{5/4}}, \end{aligned}$$

while, for $|j|\ge 2$,

$$\begin{aligned}&\sup \limits _{x\in \Xi _c, y\in [0,1]}|T_j(x,y)| \le \frac{|\log \alpha |(\mathfrak {F}_0+\mathfrak {F}_c)\left( 1+\mathfrak {F}_c^2 \right) ^{1/4} \alpha ^{|j|/2}}{\left( 1- \mathfrak {I}_c^2 \right) (\alpha -\alpha ^{|j|})},\\&\sup \limits _{x\in [0,1], y\in \Xi _c}|T_j(x,y)| \le \frac{2|\log \alpha |\,\mathfrak {F}_c\left( 1+\mathfrak {F}_c^2 \right) ^{1/4} \alpha ^{|j|/2}}{\left( 1- \mathfrak {I}_c^2 \right) ^{5/4}(\alpha -\alpha ^{|j|})}. \end{aligned}$$

From these bounds on $|T_j(x,y)|$ it is clear that the series (4.15), with $p_j$ and $q_j$ given by (4.17) and (4.18), converges absolutely and uniformly for $(x,y) \in [0,1] \times \Xi _c$ and $(x,y) \in \Xi _c \times [0,1]$, so that $\tilde{K}_t(x,\cdot )$ and $\tilde{K}_t(\cdot ,y)$ are analytic in $\Sigma _c$, for $x,y\in \mathbb {R}$, as required. The convergence of the series is also uniform in t, which implies the continuity of $t \mapsto \tilde{K}_t(\cdot ,\cdot )$. Further, the above bounds on $|T_j(x,y)|$ imply the bounds (4.22) and (4.23). $\square$

Recalling (4.19) and that the kernel of ${\tilde{K}}_t$ is continuous, we will approximate the spectrum and spectral radius of $K_t$, for $t\in \mathbb {R}$, by approximating the spectrum and spectral radius of ${\tilde{K}}_t$, considered as an operator on C[0, 1], using the Nyström method results of Sect. 3.2. Define $\tilde{K}_{t,N} :C[0,1]\rightarrow C[0,1]$, for $N\in \mathbb {N}$ and $t\in \mathbb {R}$, by [cf. (3.12)]

$$\begin{aligned} \tilde{K}_{t,N}\phi (x):= & {} J_N(\tilde{K}_t(x,\cdot )\phi )\nonumber \\ {}= & {} \frac{1}{N}\sum _{q=1}^N \tilde{K}_t(x,x_{q,N}) \phi (x_{q,N}), \quad x \in [0,1],\;\; \phi \in C[0,1], \end{aligned}$$

(4.30)

where

$$\begin{aligned} J_N\psi (x):= \sum _{q=1}^N \omega _{q,N}\phi (x_{q,N}), \quad \psi \in C[0,1],\quad N\in \mathbb {N}, \end{aligned}$$

with $x_{q,N}:= \frac{1}{N}(q-\frac{1}{2})$, $\omega _{q,N}:= N^{-1}$, for $q=1,\ldots ,N$, $N\in \mathbb {N}$. Note that $J_N\psi$ is just the N-point midpoint rule approximation to $J\psi := \int _0^1\psi (x)\, \textrm{d}x$, and that this numerical quadrature rule sequence satisfies (3.13) and (3.14), so that, for each $t\in \mathbb {R}$, ${\tilde{K}}_{t,N}\rightarrow {\tilde{K}}_t$ as $N\rightarrow \infty$ and $({\tilde{K}}_{t,N})_{N\in \mathbb {N}}$ is collectively compact. This implies, from the general result (3.9), that $\Vert (\tilde{K}_t - \tilde{K}_{t,N})\tilde{K}_t\Vert _{\infty }\rightarrow 0$ as $N\rightarrow \infty$. In our case this convergence is exponential (cf. [41, Example 12.11], [63, Sect. 19]).

Proposition 4.10

Let f and c be as in Proposition 4.8. Then, for every $t \in \mathbb {R}$ and $N\in \mathbb {N}$,

$$\begin{aligned} \left\Vert (\tilde{K}_t - \tilde{K}_{t,N})\tilde{K}_t\right\Vert _{\infty } \le \frac{2 e^{2\pi c}\Vert \tilde{K}_t\Vert _{0,c}\Vert \tilde{K}_t\Vert _{c,0}}{e^{2\pi Nc}-1}. \end{aligned}$$

To prove this proposition we will use the following classical result.

Theorem 4.11

(Theorem 9.28 in [40]) Let $\psi :\mathbb {R}\rightarrow \mathbb {C}$ be a 1-periodic function that can be extended to a bounded analytic function on $\Sigma _c$ for some $c > 0$. Then

$$\begin{aligned} \left|J\psi -J_N\psi \right|=\left|\int _0^1 \psi (x) \, \textrm{d}x - \frac{1}{N}\sum \limits _{q = 1}^N \psi (x_{q,N})\right| \le \frac{2\left\Vert \psi \right\Vert _c}{e^{2\pi N c}-1}, \quad N\in \mathbb {N}. \end{aligned}$$

Proof

It is clear from the definitions of ${\tilde{K}}_t$ and ${\tilde{K}}_{t,N}$ that it is enough to consider the case $t\in [-\pi ,\pi ]$. Assuming $t\in [-\pi ,\pi ]$, let $M_{g_t} :C[0,1] \rightarrow C[0,1]$ be multiplication by $g_t(x):= e^{ixt}$. Then $M_{g_t}$ is a surjective isometry with $M_{g_t}^{-1} = M_{g_{-t}}$. Let $\tilde{L}_t:= M_{g_t}\tilde{K}_tM_{g_t}^{-1}$ and $\tilde{L}_{t,N}:= M_{g_t}\tilde{K}_{t,N}M_{g_t}^{-1}$. Then $\tilde{L}_t$ is an integral operator with kernel $\tilde{L}_t(x,y) = e^{i(x-y)t}\tilde{K}_t(x,y)$. As $\tilde{K}_t(x+1,y) = e^{-it}\tilde{K}_t(x,y)$ and $\tilde{K}_t(x,y+1) = e^{it}\tilde{K}_t(x,y)$, we get $\tilde{L}_t(x+1,y) = \tilde{L}_t(x,y)= \tilde{L}_t(x,y+1)$ for all $x,y \in \mathbb {R}$. Moreover, $\Vert \tilde{L}_t\Vert _{0,c} \le e^{\pi c}\Vert \tilde{K}_t\Vert _{0,c}$ and $\Vert \tilde{L}_t\Vert _{c,0} \le e^{\pi c}\Vert \tilde{K}_t\Vert _{c,0}$. For $\phi \in C[0,1]$,

$$\begin{aligned} \psi (x):= \int _0^1 \tilde{L}_t(x,y)\phi (y) \, \textrm{d}y, \quad x\in \Sigma _c, \end{aligned}$$

is an analytic and 1-periodic extension of ${\tilde{L}}_t\phi$ from [0, 1] to $\Sigma _c$. Thus also $y \mapsto \tilde{L}_t(x,y)\psi (y)$ is analytic and 1-periodic for every $x \in \mathbb {R}$. By Theorem 4.11 we obtain that, for $x\in \mathbb {R}$,

$$\begin{aligned} |({\tilde{L}}_t-{\tilde{L}}_{t,N}){\tilde{L}}_t\phi (x)|= & {} |\tilde{L}_t\psi (x) - \tilde{L}_{t,N}\psi (x)| \\= & {} |J({\tilde{L}}_t(x,\cdot ) \psi ))- J_N({\tilde{L}}_t(x,\cdot ) \psi ))|\\\le & {} \frac{2 C_t(x)}{e^{2\pi Nc}-1}, \end{aligned}$$

where $C_t(x):= \sup _{y \in \Sigma _c} |\tilde{L}_t(x,y)\psi (y)|$. As $\left|\psi (y)\right| \le \sup _{z \in [0,1]} |\tilde{L}_t(y,z)|\left\Vert \phi \right\Vert _{\infty }$, for $y\in \Sigma _c$, we get

$$\begin{aligned} C_t(x) \le \Vert \tilde{L}_t\Vert _{0,c}\Vert \tilde{L}_t\Vert _{c,0}\left\Vert \phi \right\Vert _{\infty } \le e^{2\pi c}\Vert \tilde{K}_t\Vert _{0,c}\Vert \tilde{K}_t\Vert _{c,0}\left\Vert \phi \right\Vert _{\infty }, \end{aligned}$$

for all $x \in [0,1]$ and $\phi \in C[0,1]$. Therefore,

$$\begin{aligned} \left\Vert (\tilde{K}_t - \tilde{K}_{t,N})\tilde{K}_t\right\Vert _{\infty } = \left\Vert (\tilde{L}_t - \tilde{L}_{t,N})\tilde{L}_t\right\Vert _{\infty } \le \frac{2 e^{2\pi c}\Vert \tilde{K}_t\Vert _{0,c}\Vert \tilde{K}_t\Vert _{c,0}}{e^{2\pi Nc}-1}. \end{aligned}$$

$\square$

Let $A_{t,N}$ be the $N \times N$-matrix defined by [cf. (3.15)]

$$\begin{aligned} A_{t,N}(p,q) := \omega _{q,N} \tilde{K}_t(x_{p,N},x_{q,N}) = \frac{1}{N}\tilde{K}_t(x_{p,N},x_{q,N}), \quad p,q = 1,\ldots ,N. \end{aligned}$$

(4.31)

$\tilde{K}_t(\cdot ,\cdot )$ is defined by the series (4.15) which we truncate to evaluate numerically. For $M \in \mathbb {N}$ let

$$\begin{aligned} {\tilde{K}}_t^M(x,y) := \frac{1}{2\pi }\sum \limits _{j = -M}^{M} e^{ijt}\,\frac{p_j(x,y)}{1+q_j(x,y)^2}\left( \frac{1+f'(\alpha ^{x})^2}{1+f'(\alpha ^y)^2}\right) ^{1/4} \alpha ^{\frac{x+y+j}{2}}\left|\log \alpha \right|, \qquad x,y\in [0,1], \end{aligned}$$

(4.32)

where $p_j$ and $q_j$ are defined by (4.16), and consider the matrices $A_{t,N}^M$ given by $A_{t,N}^M(p,q):= \frac{1}{N}{\tilde{K}}_t^M(x_{p,N},x_{q,N})$, $p,q = 1,\ldots ,N$, so that $A_{t,N}^M$ is an approximation to $A_{t,N}$ obtained by using finitely many terms in the series defining ${\tilde{K}}_t(\cdot ,\cdot )$. Similarly, define ${\tilde{K}}^M_{t,N}$ by (4.30) with ${\tilde{K}}_{t,N}$ and ${\tilde{K}}_t(\cdot ,\cdot )$ replaced by ${\tilde{K}}^M_{t,N}$ and ${\tilde{K}}^M_t(\cdot ,\cdot )$, respectively. For $M,N\in \mathbb {N}$, with $M\ge 2$, and $t\in \mathbb {R}$, by (3.17),

$$\begin{aligned} \left\Vert A_{t,N} - A_{t,N}^M\right\Vert _{\infty } \le \Vert {\tilde{K}}_{t,N}-{\tilde{K}}_{t,N}^M\Vert _\infty \le \sup _{x,y\in [0,1]}|{\tilde{K}}_t(x,y)-{\tilde{K}}^M_t(x,y)| \le C_1(M), \end{aligned}$$

(4.33)

where, using the notations of (4.20) and Remark 4.9,

$$\begin{aligned} C_1(M):=\frac{2|\log \alpha |\mathfrak {F}_0}{\pi }\left( 1+\mathfrak {F}_0^2\right) ^{1/4} \, \mathcal {B}_M(\alpha ); \end{aligned}$$

(4.34)

this bound (4.33)–(4.34) [cf. (4.22)] is obtained as in Proposition 4.8 (set $c = 0$, only take the terms with $|j| \ge M+1$, and note that $\mathfrak {I}_0=0$), and by using (4.24). Notice that, by (4.26), $C_1(M)=O(\alpha ^{M/2})$ as $M\rightarrow \infty$.

Our aim now is to estimate $\rho (D_\Gamma ;L^2(\Gamma ))$ and $\sigma (D_\Gamma ;L^2(\Gamma ))$ by computing $\rho (A^M_{t,N})$ and $\sigma (A^M_{t,N})$ for only finitely many t. For this purpose bounds on the Lipschitz constants of $A^M_{t,N}$ and ${\tilde{K}}_{t,N}$ as functions of t will be helpful. It follows from (4.32) that, for $M,N\in \mathbb {N}$, $t\in \mathbb {R}$, and $p,q=1,\ldots ,N$,

$$\begin{aligned} \left|\frac{\partial }{\partial t} A_{t,N}^M(p,q)\right|\le & {} B_N^M(p,q) \nonumber \\:= & {} \frac{1}{2\pi N}\sum \limits _{j = -M}^{M} \frac{|j|\,\left| p_j(x_{p,N},x_{q,N})\right| }{1+q_j(x_{p,N},x_{q,N})^2}\nonumber \\ {}{} & {} \left( \frac{1+f'(\alpha ^{x_{p,N}})^2}{1+f'(\alpha ^{x_{q,N}})^2}\right) ^{1/4} \alpha ^{\frac{x_{p,N}+x_{q,N}+j}{2}}\left|\log \alpha \right|, \end{aligned}$$

(4.35)

so that, for $s,t\in \mathbb {R}$, $M,N\in \mathbb {N}$,

$$\begin{aligned} \left\Vert A^M_{t,N}-A^M_{s,N}\right\Vert _\infty \le |s-t| \left\Vert B^M_N\right\Vert _\infty . \end{aligned}$$

(4.36)

Note that, for all M and N,

$$\begin{aligned} \Vert B_N^M\Vert _\infty \le C_2 := \frac{1}{2\pi }\sup _{x,y\in [0,1]}\sum \limits _{j = -\infty }^{\infty } |j|\,\left| p_j(x,y)\right| \left( 1+\Vert f'\Vert _\infty ^2\right) ^{1/4}\alpha ^{\frac{x+y+j}{2}}\left|\log \alpha \right|, \end{aligned}$$

(4.37)

which is finite by the bounds on $p_j$ in the proof of Proposition 4.8. Similarly, from (4.15), for $t\in \mathbb {R}$, $N\in \mathbb {N}$, $x,y\in [0,1]$,

$$\begin{aligned} \left|\frac{\partial }{\partial t} {\tilde{K}}_{t,N}(x,y)\right| \le C_2 \quad \text{ so } \text{ that } \quad \Vert {\tilde{K}}_{t,N}-{\tilde{K}}_{s,N}\Vert _\infty \le |s-t| C_2, \end{aligned}$$

(4.38)

for $s,t\in \mathbb {R}$ and $N\in \mathbb {N}$, by (3.17). Further, by (3.16), (4.23) with $c=0$, and (4.25) with¹⁵$n=10$,

$$\begin{aligned} \Vert {\tilde{K}}_{t,N}\Vert _\infty \le C_3 := \frac{\left( 1+\mathfrak {F}_0^2\right) ^{1/4}}{\pi }\Bigg [ \mathfrak {G}_0\, \frac{1+\alpha ^{1/2}+\alpha ^{-1/2}}{4\alpha ^2} + 2|\log \alpha |\,\mathfrak {F}_0 \, \mathcal {B}_{10}^*(\alpha )\Bigg ], \end{aligned}$$

(4.39)

for $t\in \mathbb {R}$ and $N\in \mathbb {N}$. Provided c is such that the conditions of Lemma 4.7(iii) and (4.21) hold, we have also, by Propositions 4.8 and 4.10 and (4.25), that,

$$\begin{aligned} \left\Vert (\tilde{K}_t - \tilde{K}_{t,N})\tilde{K}_t\right\Vert _{\infty } \le \frac{C_4}{e^{2\pi Nc}-1}, \quad t\in \mathbb {R}, \;\; N\in \mathbb {N}, \end{aligned}$$

(4.40)

where $C_4:= 2e^{2\pi c}C_5C_6$, and $C_5$ and $C_6$ denote the right hand sides of (4.22) and (4.23), respectively, with the sum replaced by its upper bound $\mathcal {B}^*_{10}(\alpha )$.

As noted above Proposition 4.10, $\{{\tilde{K}}_{t,N}:N\in \mathbb {N}\}$ is collectively compact for each $t\in \mathbb {R}$. We will need, in the proof of Theorem 4.13, the following stronger statement.

Lemma 4.12

The sets $\{{\tilde{K}}_{t,N}:t\in \mathbb {R},\,N\in \mathbb {N}\}$ and $\{{\tilde{K}}^M_{t,N}:t\in \mathbb {R},\,M,N\in \mathbb {N}\}$ are collectively compact.

Proof

Where c is such that the conditions of Lemma 4.7(iii) and (4.21) hold, Proposition 4.8 implies that, for $t\in \mathbb {R}$, $N\in \mathbb {N}$, and $\phi \in C[0,1]$ with $\Vert \phi \Vert _\infty \le 1$, ${\tilde{K}}_{t,N}\phi$ has an analytic continuation from [0, 1] to $\Sigma _c$. Where $\mathcal {F}$ denotes this set of analytic continuations, it follows from (4.22) that $\mathcal {F}$ is uniformly bounded on $\Sigma _c$, so that (e.g., [58, Theorem 14.6]) $\mathcal {F}$ is normal, so that $\{{\tilde{K}}_{t,N}\phi :t\in \mathbb {R}, N\in \mathbb {N}, \Vert \phi \Vert _\infty \le 1\}$ is relatively compact, i.e., $\{{\tilde{K}}_{t,N}:t\in \mathbb {R},\,N\in \mathbb {N}\}$ is collectively compact. The same argument applies to the family ${\tilde{K}}^M_{t,N}$, on noting, by inspection of the proof of Proposition 4.8, that, under the same conditions on c, ${\tilde{K}}_t^M(\cdot ,y)$ extends to an analytic function on $\Sigma _c$ bounded by the right hand side of (4.22), for all $t,y\in \mathbb {R}$, $M\in \mathbb {N}$. $\square$

In the following result, which holds for every $\Gamma$ that satisfies our standing assumption (4.8), we propose an approximation for the spectrum of $D_\Gamma$ as the union of the spectra of finitely many finite matrices, and show that this approximation converges in the Hausdorff metric. In this theorem $\sigma (\tilde{K}^M_{t,N})$ denotes the spectrum of ${\tilde{K}}^M_{t,N}$ either on $L^2(0,1)$ or on C[0, 1] [cf. (4.19)], which coincides with $\{0\}\cup \sigma (A^M_{t,N})$ by Lemma 3.7(ii), and $\sigma (D_\Gamma )$ denotes the spectrum of $D_\Gamma$ on $L^2(\Gamma )$, which coincides with the essential spectrum by Corollary 4.2.

Theorem 4.13

Choose sequences $(m_N)_{N\in \mathbb {N}}, (M_N)_{N\in \mathbb {N}}\subset \mathbb {N}$ such that $m_N, M_N\rightarrow \infty$ as $N\rightarrow \infty$, and for each N, let $T_N:=\{\pm (k-1/2)\pi /m_N:k=1,\ldots ,m_N\}$. Then

(4.41)

as $N\rightarrow \infty$.

Proof

Set $\Lambda := \sigma (D_{\Gamma })$, so that $\Lambda = \bigcup _{t \in [-\pi ,\pi ]} \sigma (\tilde{K}_t)$ by Theorem 4.4 and (4.19), and set $\Lambda _N:= \sigma ^N(D_\Gamma )$. We first observe that $(\Lambda _N)_{N \in \mathbb {N}}$ is uniformly bounded as $\{\Vert \tilde{K}^{M_N}_{t,N}\Vert _{\infty }:N\in \mathbb {N}, \,t\in T_N\}$ is bounded, by (4.33) and (4.39). Thus, as noted above Theorem 3.6, to prove that it is enough to show that $\lim \inf \Lambda _N = \lim \sup \Lambda _N = \Lambda$.

Next, we note that if the sequences $(\mathfrak {M}_k)_{k\in \mathbb {N}}\subset \mathbb {N}$, $(N_k)_{k\in \mathbb {N}}\subset \mathbb {N}$, and $(t_k)_{k\in \mathbb {N}}\subset [-\pi ,\pi ]$ satisfy $N_k\rightarrow \infty$, $\mathfrak {M}_k\rightarrow \infty$, $t_k\rightarrow t\in [-\pi ,\pi ]$, as $k\rightarrow \infty$, then the sequence $(\tilde{K}^{\mathfrak {M}_k}_{t_k,N_k})_{k \in \mathbb {N}}$ is collectively compact by Lemma 4.12, and converges strongly to $\tilde{K}_t$ by (4.38) and (4.33), and since ${\tilde{K}}_{t,N_k}\rightarrow {\tilde{K}}_t$, as noted above Proposition 4.10. Thus also , by Theorem 3.6.

For every $t\in [-\pi ,\pi ]$ we can choose a sequence $(t_N)_{N\in \mathbb {N}}$ such that $t_N\in T_N$ for each N and $t_N\rightarrow t$. Then, by the observation just made, as $N\rightarrow \infty$, so that $\lim \inf \Lambda _N \supset \lim \inf \sigma (\tilde{K}^{M_N}_{t_N,N}) = \sigma ({\tilde{K}}_t)$. Thus $\lim \inf \Lambda _N \supset \Lambda$.

If $\lambda \in \lim \sup \Lambda _N$ then there exists a sequence $(N_k)_{k\in \mathbb {N}}\subset \mathbb {N}$ and a sequence $(\lambda _k)_{k\in \mathbb {N}}$ such that $\lambda _k\rightarrow \lambda$ and, for each k, $\lambda _k\in \Lambda _{N_k}$, so that $\lambda _k\in \sigma (\tilde{K}^{\mathfrak {M}_k}_{t_k,N_k})$, for some $t_k\in T_{N_k}$, where $\mathfrak {M}_k:= M_{N_k}$. By passing to a subsequence if necessary, we may assume that $t_k\rightarrow t\in [-\pi ,\pi ]$, so that , in particular $\lim \inf \sigma (\tilde{K}^{\mathfrak {M}_k}_{t_k,N_k}) = \sigma ({\tilde{K}}_t)$, so that $\lambda \in \sigma ({\tilde{K}}_t)\subset \Lambda$. Thus $\lim \sup \Lambda _N\subset \Lambda \subset \lim \inf \Lambda _N$. But also $\lim \inf \Lambda _N \subset \lim \sup \Lambda _N$, so the proof is complete. $\square$

Remark 4.14

(Reduced computation expression for $\sigma ^N(D_\Gamma )$) Note that, by Remark 4.5 and Lemma 3.7(ii), $\sigma ^N(D_\Gamma )$, given by (4.41), can be written as

$$\begin{aligned} \sigma ^N(D_\Gamma ) = \{0\} \cup \{\lambda , \overline{\lambda }: \lambda \in \sigma (A^{M_N}_{t,N}),\, t\in T_N, \, t>0\}. \end{aligned}$$

Our second main result, obtained by applying Theorem 3.8,¹⁶ provides a criterion, given $\rho _0>0$, for $\rho (D_\Gamma ;L^2(\Gamma ))<\rho _0$. Note that this criterion requires computation of spectral quantities only for the $N\times N$ matrices $A_{t,N}^M$ for finitely many $t\in [0,\pi ]$.

Theorem 4.15

Under our standing assumption (4.8), define g by (4.10) so that, by Lemma 4.7, g has an analytic continuation to $\Sigma _c$, for some $c>0$, such that g, $g^\prime$, and $g^{\prime \prime }$ are bounded on $\Sigma _c$, and, without loss of generality, assume that (4.21) holds. Suppose that $\rho _0>0$, $m,M,N\in \mathbb {N}$, with $M\ge 2$. For $k=1,\ldots ,m$, set $t_k:= (k-1/2)\pi /m$, suppose that $\rho (A^M_{t_k,N})<\rho _0$ for $k=1,\ldots ,m$, and recursively define $\mu _{k,\ell }$, for $\ell =1,2,\ldots$, by $\mu _{k,1}:= \rho _0$, and by

$$\begin{aligned} \nu _{k,\ell } := \left\Vert (A_{t_k,N}^M - \mu _{k,\ell } I)^{-1}\right\Vert _{\infty }^{-1} \quad \textrm{and} \quad \mu _{k,\ell +1} := \mu _{k,\ell }e^{i\frac{\nu _{k,\ell }}{2\rho _0}}, \quad \textrm{for }\, \ell \in \mathbb {N}. \end{aligned}$$

(4.42)

Further, for $k=1,\ldots ,m$, let $n_k$ denote the smallest integer such that $\sum _{\ell =1}^{n_k} \nu _{k,\ell } \ge 4\pi \rho _0$, and let

$$\begin{aligned} R_{m,M,N}:=\min _{{\mathop {\ell =1,\ldots ,n_k}\limits ^{k=1,\ldots ,m}}} \left( \frac{\nu _{k,\ell }}{4}+\frac{\nu _{k,\ell +1}}{2}\right) . \end{aligned}$$

If

$$\begin{aligned} \mathcal {L}_c(f,N):= & {} \frac{C_4}{e^{2\pi Nc}-1} < \mathcal {R}_c(f,m,M,N) \nonumber \\:= & {} r^2_0\left( 1 + C_3\left( R_{m,M,N} - C_1(M)- \frac{\pi }{2m}\Vert B_N^M\Vert _\infty \right) ^{-1} \right) ^{-1}, \end{aligned}$$

(4.43)

then $\rho (D_\Gamma ;L^2(\Gamma )) < \rho _0$. Conversely, if $\rho (D_\Gamma ;L^2(\Gamma )) < \rho _0$, then, provided m and M are sufficiently large, there exists $N_0\in \mathbb {N}$ such that (4.43) holds and $\rho (A_{t,N}^M)<\rho _0$ for $t\in [0,\pi ]$ and all $N\ge N_0$.

Proof

By Theorem 4.4, Remark 4.5, and (4.19), to show that $\rho (D_\Gamma ;L^2(\Gamma )) < \rho _0$, it is enough to check that $\rho ({\tilde{K}}_t; {C[0,1]})<\rho _0$ for every $t\in [0,\pi ]$. So pick $t\in [0,\pi ]$. Then $|t-t_k| \le \pi /(2\,m)$, for some $k\in \{1,\ldots ,m\}$. To conclude that $\rho ({\tilde{K}}_t; {C[0,1]})<\rho _0$ we apply Theorem 3.8 with $K={\tilde{K}}_t$, $K_N={\tilde{K}}_{t,N}$, $A_N=A_{t,N}$, and $A_N^{\dag }=A_{t_k,N}^M$, noting that, with these definitions, $\Vert K_N\Vert _\infty = \Vert {\tilde{K}}_{t,N}\Vert _\infty \le C_3$,

$$\begin{aligned} \Vert A_N-A_N^{\dag }\Vert _\infty \le \Vert A_{t,N}-A_{t,N}^M\Vert _\infty + \Vert A^M_{t,N}-A_{t_k,N}^M\Vert _\infty \le C_1(M) + \frac{\pi }{2m}\Vert B_N^M\Vert _\infty , \end{aligned}$$

by (4.33) and (4.36), and $\Vert (K-K_N)K\Vert _\infty \le C_4/(e^{2\pi Nc}-1)$, by (4.40).

Conversely, if $\rho (D_\Gamma ;L^2(\Gamma )) < \rho _0$ then, by Theorem 4.4 and (4.19), $\rho ({\tilde{K}}_t; {C[0,1]})<\rho _0$ for $t\in [0,\pi ]$. Arguing as in the proof of Theorem 3.8, this implies that for every $t\in [0,\pi ]$ there exists $N_0(t)$ such that $\Vert ({\tilde{K}}_{t,N}-\lambda I)^{-1}\Vert$ is bounded uniformly in $\lambda$ and N for $N\ge N_0(t)$ and $|\lambda |\ge \rho _0$. This, combined with the estimate (4.38) and a standard compactness argument, implies that, for some $N^*\in \mathbb {N}$, and $c^*>0$, $\Vert ({\tilde{K}}_{t,N}-\lambda I)^{-1}\Vert ^{-1}\ge c^*$ for all $N\ge N^*$, $|\lambda |\ge \rho _0$, and $t\in [0,\pi ]$. It follows, by (4.33) and (3.1), and recalling Lemma 3.7(ii), that, for all $t\in [0,\pi ]$ and all sufficiently large M and N, $\rho (A_{t,N}^M)=\rho ({\tilde{K}}^M_{t,N}; {C[0,1]})< \rho _0$ and $\Vert ({\tilde{K}}^M_{t,N}-\lambda I)^{-1}\Vert ^{-1}\ge c^*/2$, for $|\lambda |\ge \rho _0$. Thus, and by Lemma 3.7(iii), $R_{m,N,M}\ge 3c^*/8$, for all $m\in \mathbb {N}$ and all sufficiently large M and N. It follows that (4.43) holds, for all sufficiently large m, M, and N, since (4.34) and (4.26), and that $\Vert B_N^M\Vert _\infty$ is bounded independently of M and N by (4.37), imply that the right hand side of (4.43) is positive and bounded away from zero. $\square$

4.3 The 2D case: two-sided infinite graphs

We now extend the results of the previous subsection, for the case when $d=2$ and $A=\mathbb {R}_+$, to the more involved case¹⁷ where $d=2$ and $A=\mathbb {R}$. Our goals and methods are those of Sect. 4.2, but, where $\dot{\mathbb {R}}:= \mathbb {R}{\setminus } \{0\}$, we assume now that $f\in \mathcal {A}(\dot{\mathbb {R}})$, the space of functions $\mathbb {R}\rightarrow \mathbb {R}$ that are real analytic on $\dot{\mathbb {R}}$ with $f(0)=0$ (a prototypical example is Fig. 3). Thus our standing assumption through this subsection is that

$$\begin{aligned} \Gamma= & {} \left\{ (x,f(x)): x \in \mathbb {R}\right\} \; \text{ where } \; f\in \mathcal {A}(\dot{\mathbb {R}}) \; \text{ and, } \text{ for } \text{ some }\,\nonumber \\{} & {} \alpha \in (0,1), \; f(\alpha x) = \alpha f(x), \quad x\in \mathbb {R}. \end{aligned}$$

(4.44)

Fig. 3

Graph of $f :\mathbb {R}\rightarrow \mathbb {R}$, $f(x):= \left|x\right|\sin ^2(\pi \log _{\alpha }\left|x\right|)$, in the case $\alpha = \frac{2}{3}$

Define $f_\pm :\mathbb {R}_+\rightarrow \mathbb {R}$ by $f_\pm (x):= f(\pm x)$, $x\in \mathbb {R}_+$, and $g_\pm :\mathbb {R}\rightarrow \mathbb {R}$ [cf. (4.10)] by

$$\begin{aligned} g_\pm (x) := \alpha ^{-x}f_\pm (\alpha ^x), \quad x\in \mathbb {R}, \quad \text{ so } \text{ that } \quad f_\pm (x) = xg_\pm (\log _\alpha x), \quad x\in \mathbb {R}_+. \end{aligned}$$

(4.45)

Note that the assumption $f\in \mathcal {A}(\dot{\mathbb {R}})$ is equivalent to the assumption that $f_\pm \in \mathcal {A}(\mathbb {R}_+)$ and $f(0)=0$. Note also that Lemma 4.7 applies with f and g replaced by $f_\pm$ and $g_\pm$. One simple consequence of these observations [see the discussion below (4.11)] is that (4.44) implies $f\in C^{0,1}(\mathbb {R})$.

As in the one-sided case, our starting point is the formula for $\sigma (D_\Gamma )$ in Corollary 4.4 in terms of $\sigma (K_t)$, for $t\in [-\pi ,\pi ]$. Again, $K_t$ is compact for every $t\in \mathbb {R}$ by Corollary 4.6, and, following the pattern of Sect. 4.2, our goal is to compute $\sigma (K_t)$ and $\rho (K_t)$ by the results of Sect. 3.2. As in Sect. 4.2, to make use of these results it is convenient to work with integral operators on [0, 1] rather than $\Gamma _0$. Reflecting that $\Gamma _0$ has two components, $\Gamma _0^+:= \{(x,f(x))\in \Gamma _0:x>0\}$ and $\Gamma _0^-:= \{(x,f(x))\in \Gamma _0:x<0\}$, it is natural to work in this case with a unitary operator $U:L^2(\Gamma _0)\rightarrow (L^2(0,1))^2$, defined by $U\psi := (U_+P_+\psi , U_-P_-\psi )^T$, $\psi \in L^2(\Gamma _0)$, where $P_\pm$ is restriction to $\Gamma _0^\pm$, i.e. $P_\pm \psi := \psi |_{\Gamma _0^\pm }$, $\psi \in L^2(\Gamma _0)$, and $U_\pm :L^2(\Gamma _0^\pm )\rightarrow L^2(0,1)$ is the unitary operator defined by [cf. (4.12)]

$$\begin{aligned}{} & {} U_\pm \phi (x):= \phi (\pm \alpha ^{x}, f_\pm (\alpha ^x)) (1+f_\pm '(\alpha ^x)^2)^{1/4} \alpha ^{x/2}|\log \alpha |^{1/2},\\ {}{} & {} \qquad x\in [0,1], \;\; \phi \in L^2(\Gamma _0^\pm ). \end{aligned}$$

With U as given above we define an operator ${\tilde{K}}_t$ on $(L^2(0,1))^2$, which is unitarily equivalent to $K_t$, by (4.13). It is easy to see [cf. (4.14)] that

$$\begin{aligned} {\tilde{K}}_t = \begin{pmatrix} {\tilde{K}}^-_t &{} {\tilde{L}}^-_t\\ {\tilde{L}}^+_t &{} {\tilde{K}}^+_t \end{pmatrix}, \quad t\in \mathbb {R}, \end{aligned}$$

(4.46)

where the entries of ${\tilde{K}}_t$ are integral operators on $L^2(0,1)$ with continuous kernels. The kernels of ${\tilde{K}}_t^\pm$ are ${\tilde{K}}^\pm _t(\cdot ,\cdot )$, where [cf. (4.15)]

$$\begin{aligned} \tilde{K}^\pm _t(x,y) := \frac{1}{2\pi }\sum \limits _{j = -\infty }^{\infty } e^{ijt}\,\frac{p^\pm _j(x,y)}{1+q^\pm _j(x,y)^2}\left( \frac{1+f_\pm '(\alpha ^{x})^2}{1+f_\pm '(\alpha ^y)^2}\right) ^{1/4} \alpha ^{\frac{x+y+j}{2}}\left|\log \alpha \right|, \quad x,y\in \mathbb {R}, \end{aligned}$$

(4.47)

and $p_j^\pm$ and $q_j^\pm$ are defined by (4.16), (4.17), and (4.18), but with f replaced by $f_\pm$. The kernels of ${\tilde{L}}_t^\pm$ are ${\tilde{L}}_t^\pm (\cdot ,\cdot )$, where, for $x,y\in \mathbb {R}$,

$$\begin{aligned} \tilde{L}^\pm _t(x,y) := \frac{1}{2\pi }\sum \limits _{j = -\infty }^{\infty } e^{ijt}\, \frac{{\tilde{p}}^\pm _j(x,y)}{1+{\tilde{q}}^\pm _j(x,y)^2}\left( \frac{1+f_\pm '(\alpha ^{x})^2}{1+f_\mp '(\alpha ^y)^2}\right) ^{1/4}\alpha ^{\frac{x+y+j}{2}}\left|\log \alpha \right| \end{aligned}$$

(4.48)

and

$$\begin{aligned}{} & {} {\tilde{p}}^\pm _j(x,y) := \frac{(\alpha ^y+\alpha ^{x+j})f_\mp ^\prime (\alpha ^y) + f_\pm (\alpha ^{x+j}) - f_\mp (\alpha ^y)}{(\alpha ^{x+j}+\alpha ^y)^2}, \nonumber \\{} & {} {\tilde{q}}^\pm _j(x,y) := \frac{f_\pm (\alpha ^{x+j}) -f_\mp (\alpha ^y)}{\alpha ^{x+j}+\alpha ^y}. \end{aligned}$$

(4.49)

Analogously to (4.19), we have that

$$\begin{aligned} \sigma (K_t;L^2(\Gamma _0)) = \sigma ({\tilde{K}}_t;(L^2(0,1))^2)=\sigma ({\tilde{K}}_t;(C[0,1])^2), \quad t\in \mathbb {R}. \end{aligned}$$

(4.50)

As in the one-sided case [see the discussion around (4.30)], to estimate spectral properties of ${\tilde{K}}_t$ as an operator on $(C[0,1])^2$ we approximate ${\tilde{K}}^\pm _t$ by finite rank operators ${\tilde{K}}^\pm _{t,N}$ given by

$$\begin{aligned} \tilde{K}^\pm _{t,N}\phi (x):= \frac{1}{N}\sum _{n=1}^N \tilde{K}^\pm _t(x,x_{n,N}) \phi (x_{n,N}), \quad \phi \in C[0,1], \quad x \in [0,1], \quad N\in \mathbb {N}. \end{aligned}$$

Similarly, we approximate $\tilde{L}^\pm _{t}$ by $\tilde{L}^\pm _{t,N}$, leading to finite rank approximations ${\tilde{K}}_{t,N}$, $N\in \mathbb {N}$, to ${\tilde{K}}_t$, given by (4.46) with ${\tilde{K}}^\pm _t$ and ${\tilde{L}}^\pm _t$ replaced by ${\tilde{K}}^\pm _{t,N}$ and ${\tilde{L}}^\pm _{t,N}$. Arguing as below (4.30), we have that ${\tilde{K}}_{t,N}\rightarrow {\tilde{K}}_t$ and $({\tilde{K}}_{t,N})_{N\in \mathbb {N}}$ is collectively compact, so that $\Vert ({\tilde{K}}_t-{\tilde{K}}_{t,N}){\tilde{K}}_t\Vert _\infty \rightarrow 0$ as $N\rightarrow \infty$, where $\Vert \cdot \Vert _\infty$ here denotes the operator norm of an operator on $(C[0,1])^2$ equipped with the norm $\Vert \cdot \Vert _\infty$ defined by $\Vert \phi \Vert _\infty :=\max (\Vert \phi _+\Vert _\infty ,\Vert \phi _-\Vert _\infty )$, for $\phi =(\phi _+,\phi _-)^T\in (C[0,1])^2$ Indeed, we have the following analogue of Proposition 4.10, in which our other norm notations are as defined above Proposition 4.8.

Proposition 4.16

Let $t\in \mathbb {R}$ and $c>0$ and assume that, for every $x,y\in \mathbb {R}$, ${\tilde{K}}^{\pm }_t(x,\cdot )$, ${\tilde{L}}^{\pm }_t(x,\cdot )$, ${\tilde{K}}^{\pm }_t(\cdot ,y)$ and ${\tilde{L}}^{\pm }_t(\cdot ,y)$ have analytic continuations from $\mathbb {R}$ to $\Sigma _c$ that are bounded in $\Sigma _c$. Then $\Vert (\tilde{K}_t - \tilde{K}_{t,N})\tilde{K}_t\Vert _{\infty } \le 2e^{2\pi c}C^*/(e^{2\pi N c}-1)$, where

$$\begin{aligned} C^*&:= \max \bigg \{ \Vert {\tilde{K}}^-_t\Vert _{0,c} \Vert {\tilde{K}}^-_t\Vert _{c,0} + \Vert {\tilde{L}}^-_t\Vert _{0,c} \Vert {\tilde{L}}^+_t\Vert _{c,0} + \Vert {\tilde{K}}^-_t\Vert _{0,c} \Vert {\tilde{L}}^-_t\Vert _{c,0} + \Vert {\tilde{L}}^-_t\Vert _{0,c} \Vert {\tilde{K}}^+_t\Vert _{c,0},\nonumber \\&\quad \Vert {\tilde{L}}^+_t\Vert _{0,c} \Vert {\tilde{K}}^-_t\Vert _{c,0} + \Vert {\tilde{K}}^+_t\Vert _{0,c} \Vert {\tilde{L}}^+_t\Vert _{c,0} + \Vert {\tilde{L}}^+_t\Vert _{0,c} \Vert {\tilde{L}}^-_t\Vert _{c,0} + \Vert {\tilde{K}}^+_t\Vert _{0,c} \Vert {\tilde{K}}^+_t\Vert _{c,0} \bigg \}. \end{aligned}$$

(4.51)

Proof

Notice first that

$$\begin{aligned} ({\tilde{K}}_t-K^N_t){\tilde{K}}_t =\begin{pmatrix} ({\tilde{K}}^-_t - K^-_{t,N}){\tilde{K}}^-_t+({\tilde{L}}^-_t-L^-_{t,N}){\tilde{L}}^+_t &{} ({\tilde{K}}^-_t - K^-_{t,N}){\tilde{L}}^-_t+({\tilde{L}}^-_t-L^-_{t,N}){\tilde{K}}^+_t\\ ({\tilde{L}}^+_t-L^+_{t,N}){\tilde{K}}^-_t + ({\tilde{K}}^+_t-K^+_{t,N}){\tilde{L}}^+_t &{} ({\tilde{L}}^+_t-L^+_{t,N}){\tilde{L}}^-_t + ({\tilde{K}}^+_t-K^+_{t,N}){\tilde{K}}^+_t \end{pmatrix}, \end{aligned}$$

and denote the entries of this matrix by $A_{j,k}$, $j,k=1,2$. For each of these terms we obtain an estimate similar to that in Proposition 4.10, by arguing as in the proof of that proposition. Since also

$$\begin{aligned} \Vert ({\tilde{K}}_t - K^N_t){\tilde{K}}_t\Vert _{\infty } = \max _{j=1,2} \sum _{k=1}^2 \Vert A_{j,k}\Vert _{\infty }, \end{aligned}$$

the result follows. $\square$

Let us now estimate, under our standing assumption (4.44), the norms of the kernels that appear in Proposition 4.16 [cf. Proposition 4.8].

Proposition 4.17

Given (4.44), define $g_\pm$ by (4.45) so that, by Lemma 4.7, $g_\pm$ have analytic continuations to $\Sigma _c$, for some $c>0$, such that $g_\pm$, $g_\pm '$, and $g_\pm ^{''}$ are bounded on $\Sigma _c$ and $g_\pm (z+1)=g_\pm (z)$, $z\in \Sigma _c$. Let

$$\begin{aligned}&\mathfrak {I}_{c,\pm } := \Vert {{\,\textrm{Im}\,}}g_\pm \Vert _c + \Vert {{\,\textrm{Im}\,}}g_\pm ^\prime \Vert _c/|\log \alpha | \quad \textrm{and}\nonumber \\&\quad \mathfrak {K}_{c,\pm } := \alpha ^{-1}|\log \alpha |\max (\Vert g_\pm \Vert _c,\Vert g_\mp \Vert _0), \end{aligned}$$

(4.52)

and set

$$\begin{aligned}&\mathfrak {F}_{d,\pm } := \Vert g_\pm \Vert _d + \Vert g_\pm ^\prime \Vert _d/|\log \alpha | \quad \textrm{and}\nonumber \\&\quad \mathfrak {G}_{d,\pm } := \Vert g_\pm ^\prime \Vert _d + \Vert g_\pm ^{\prime \prime }\Vert _d/|\log \alpha |, \quad \textrm{for }\, d=0,c. \end{aligned}$$

(4.53)

If

$$\begin{aligned} c \le \arccos (\alpha )/\left|\log \alpha \right|, \quad \mathfrak {I}_{c,\pm }< 1,\quad \textrm{and} \quad \Vert {{\,\textrm{Im}\,}}g_\pm \Vert _c + \alpha c\mathfrak {K}_{c,\pm } < \alpha ^2, \end{aligned}$$

(4.54)

then ${\tilde{K}}^{\pm }_t(x,\cdot )$, ${\tilde{L}}^{\pm }_t(x,\cdot )$, ${\tilde{K}}^{\pm }_t(\cdot ,y)$, and ${\tilde{L}}^{\pm }_t(\cdot ,y)$ extend to bounded analytic functions on $\Sigma _c$ for all $x,y,t\in \mathbb {R}$, and

$$\begin{aligned} \Vert \tilde{K}^\pm _t\Vert _{c,0}&\le \frac{\left( 1+\mathfrak {F}_{c,\pm }^2\right) ^{1/4}}{\pi \left( 1- \mathfrak {I}_{c,\pm }^2\right) } \Bigg [ \mathfrak {G}_{c,\pm }\, \frac{1+\alpha ^{1/2}+\alpha ^{-1/2}}{4\alpha ^2} + |\log \alpha |(\mathfrak {F}_{c,\pm }+\mathfrak {F}_{0,\pm }) \, \sum _{j=2}^\infty \frac{\alpha ^{j/2}}{\alpha -\alpha ^j}\Bigg ],\nonumber \\ \Vert \tilde{K}^\pm _t\Vert _{0,c}&\le \frac{\left( 1+\mathfrak {F}_{0,\pm }^2\right) ^{1/4}}{\pi \left( 1- \mathfrak {I}_{c,\pm }^2\right) ^{5/4}} \Bigg [ \mathfrak {G}_{c,\pm }\, \frac{1+\alpha ^{1/2}+\alpha ^{-1/2}}{4\alpha ^2} + 2|\log \alpha |\,\mathfrak {F}_{c,\pm } \, \sum _{j=2}^\infty \frac{\alpha ^{j/2}}{\alpha -\alpha ^j}\Bigg ],\nonumber \\ \Vert {\tilde{L}}^{\pm }_t\Vert _{c,0}&\le \frac{1}{2\pi }\frac{\left|\log \alpha \right|\mathfrak {F}_{0,\mp } + \mathfrak {K}_{c,\pm }}{1 - \alpha ^{-4}\left( \left\Vert {{\,\textrm{Im}\,}}g_\pm \right\Vert _c + \alpha c\mathfrak {K}_{c,\pm }\right) ^2} \,\left( 1 + \mathfrak {F}_{c,\pm }^2\right) ^{1/4} \sum \limits _{j = -\infty }^{\infty } \frac{\alpha ^{j/2}}{\alpha ^{j+2}+\alpha ^2}, \end{aligned}$$

(4.55)

$$\begin{aligned} \Vert {\tilde{L}}^{\pm }_t\Vert _{0,c}&\le \frac{1}{2\pi } \frac{\left|\log \alpha \right|\mathfrak {F}_{c,\mp } + \mathfrak {K}_{c,\mp }}{1 - \alpha ^{-4}\left( \left\Vert {{\,\textrm{Im}\,}}g_\mp \right\Vert _c + \alpha c\mathfrak {K}_{c,\mp }\right) ^2}\,\left( \frac{1 + \mathfrak {F}_{0,\pm }^2}{1 - \mathfrak {I}_{c,\mp }^2}\right) ^{1/4}\sum \limits _{j = -\infty }^{\infty } \frac{\alpha ^{j/2}}{\alpha ^{j+2}+\alpha ^2}. \end{aligned}$$

(4.56)

Moreover, the mappings $\mathbb {R}\rightarrow C([0,1]\times [0,1])$, $t \mapsto \tilde{K}^{\pm }_t(\cdot ,\cdot )$ and $t \mapsto \tilde{L}^{\pm }_t(\cdot ,\cdot )$, are continuous.

Remark 4.18

For fixed $\alpha \in (0,1)$, let $F(x):= (\alpha ^{x/2} + \alpha ^{-x/2})^{-1}$, for $x\in \mathbb {R}$, so that the jth term in the sum in (4.55) and (4.56) is $\alpha ^{-2}F(j)$, and note that F is even. Arguing as in Remark 4.9, since F is decreasing on $[0,\infty )$,

$$\begin{aligned} \sum _{j=n+1}^\infty \frac{\alpha ^{j/2}}{\alpha ^{j+2}+\alpha ^2} = \alpha ^{-2}\sum _{j=n+1}^\infty F(j) \le \mathcal {C}_n(\alpha ) := \alpha ^{-2}\int _n^\infty F(x)\, \textrm{d}x = \frac{2\arctan (\alpha ^{n/2})}{\alpha ^2\left|\log \alpha \right|}, \end{aligned}$$

(4.57)

for $0<\alpha < 1$ and $n\in \mathbb {N}_0:=\mathbb {N}\cup \{0\}$. Thus, for $0<\alpha < 1$ and $n\in \mathbb {N}_0$,

$$\begin{aligned}&\sum \limits _{j = -\infty }^{\infty } \frac{\alpha ^{j/2}}{\alpha ^{j+2}+\alpha ^2} \le \mathcal {C}_n^*(\alpha ) := \frac{1}{2\alpha ^2} + 2\alpha ^{-2}\sum _{j=1}^n F(j) + 2\mathcal {C}_n(\alpha ) \quad \text{ and } \end{aligned}$$

(4.58)

$$\begin{aligned}&\left|\sum \limits _{j = -\infty }^{\infty } \frac{\alpha ^{j/2}}{\alpha ^{j+2}+\alpha ^2} - \sum \limits _{j = -n}^n \frac{\alpha ^{j/2}}{\alpha ^{j+2}+\alpha ^2}\right| \le 2 \mathcal {C}_n(\alpha ). \end{aligned}$$

(4.59)

Proof of Proposition 4.17

The results for $\tilde{K}_t^\pm (\cdot ,\cdot )$ follow immediately from Proposition 4.8, applied with f replaced by $f_\pm$. It remains to consider the off-diagonal entries ${\tilde{L}}^{\pm }_t$. We give the detail for $\tilde{L}_t^+$; the results for $\tilde{L}_t^-$ follow by the same argument with the roles of $f_+$ and $f_-$ reversed.

We extend the definition of ${\tilde{L}}^+_t$ via (4.48) to all $x,y\in \Sigma _c$. The upcoming computations will show that this is well-defined and the estimates (4.55) and (4.56) hold. Denote by $T_j(x,y)$ the jth term in the sum (4.48), that is,

$$\begin{aligned} T_j(x,y) := e^{ijt} \frac{{\tilde{p}}^+_j(x,y)}{1+{\tilde{q}}^+_j(x,y)^2}\left( \frac{1+f_+'(\alpha ^{x})^2}{1+f_-'(\alpha ^y)^2}\right) ^{1/4}\alpha ^{\frac{x+y+j}{2}}\left|\log \alpha \right|, \quad x,y\in \Sigma _c. \end{aligned}$$

(4.60)

We clearly have $T_j(x+1,y)=e^{-it}T_{j+1}(x,y)$ and the dilation invariance, $f(\alpha x) = \alpha f(x)$, implies that $T_j(x,y+1)=e^{it}T_{j-1}(x,y)$ for $x,y\in \Sigma _c$, $j \in \mathbb {Z}$. Thus, to prove the uniform convergence of the series (4.48) and the bounds (4.55) and (4.56), it suffices to restrict consideration to $x,y \in \Xi _c = \{z \in \mathbb {C}:$ ${{\,\textrm{Re}\,}}z \in [0,1]$, ${{\,\textrm{Im}\,}}z \in (-c,c)\}$. First notice that, for $x,y \in \Xi _c$,

$$\begin{aligned} \left|\alpha ^{x+j}+\alpha ^y\right|\ge & {} \alpha ^{{{\,\textrm{Re}\,}}x + j}\cos (\log \alpha {{\,\textrm{Im}\,}}x) + \alpha ^{{{\,\textrm{Re}\,}}y}\cos (\log \alpha {{\,\textrm{Im}\,}}y) \nonumber \\\ge & {} \alpha ^{{{\,\textrm{Re}\,}}x + j+1} + \alpha ^{{{\,\textrm{Re}\,}}y + 1}\ge \alpha ^{j+2} + \alpha ^2, \end{aligned}$$

(4.61)

by the first of (4.54), which implies that $\cos (t\log \alpha )\ge \alpha$, for $-c\le t\le c$. Hence, for $x,y\in \Xi _c$ and $j\in \mathbb {Z}$, recalling (4.45),

$$\begin{aligned} \left|{\tilde{q}}^+_j(x,y)\right| = \left|\frac{\alpha ^{x+j}g_+(x) - \alpha ^yg_-(y)}{\alpha ^{x+j}+\alpha ^y}\right| \le \frac{\alpha ^{{{\,\textrm{Re}\,}}x+j}|g_+(x)| + \alpha ^{{{\,\textrm{Re}\,}}y}|g_-(y)|}{\alpha ^{{{\,\textrm{Re}\,}}x + j+1}+\alpha ^{{{\,\textrm{Re}\,}}y + 1}}. \end{aligned}$$

Using the estimate

$$\begin{aligned} (a+b)/(c+d) \le \max \left\{ a/c,b/d\right\} , \end{aligned}$$

(4.62)

which holds for all $a,b\ge 0$, $c,d>0$, it follows that

$$\begin{aligned}{} & {} \sup \limits _{x \in \Xi _c,y \in [0,1]} \left|{\tilde{q}}^+_j(x,y)\right| \le \mathfrak {K}_{c,+}/|\log \alpha | \quad \text{ and } \\{} & {} \sup \limits _{x \in [0,1],y \in \Xi _c} \left|{\tilde{q}}^+_j(x,y)\right| \le \mathfrak {K}_{c,-}/|\log \alpha |, \quad j\in \mathbb {Z}. \end{aligned}$$

Noting that (4.27) holds with f and g replaced with $f_\pm$ and $g_\pm$, we have also that $|f_\pm ^\prime (\alpha ^y)|\le \mathfrak {F}_{0,\pm }$, for $y\in [0,1]$, while $|f_\pm ^\prime (\alpha ^y)|\le \mathfrak {F}_{c,\pm }$, for $y\in \Xi _c$. Thus

$$\begin{aligned} \sup \limits _{x \in \Xi _c,y \in [0,1]} \left|{\tilde{p}}^+_j(x,y)\right|&\le \sup \limits _{x \in \Xi _c,y \in [0,1]} \frac{|f_-'(\alpha ^y)| + \left|{\tilde{q}}^+_j(x,y)\right|}{\left|\alpha ^{x+j}+\alpha ^y\right|} \le \frac{\mathfrak {F}_{0,-} + \mathfrak {K}_{c,+}/|\log \alpha |}{\alpha ^{j+2}+\alpha ^2} \quad \text{ and }\\ \sup \limits _{x \in [0,1],y \in \Xi _c} \left|{\tilde{p}}^+_j(x,y)\right|&\le \frac{\mathfrak {F}_{c,-} + \mathfrak {K}_{c,-}/|\log \alpha |}{\alpha ^{j+2}+\alpha ^2}. \end{aligned}$$

Moreover, for $x,y\in \Xi _c$,

$$\begin{aligned} \left|{{\,\textrm{Im}\,}}{\tilde{q}}^+_j(x,y)\right|&= \left|{{\,\textrm{Im}\,}}\left( \frac{\alpha ^{x+j}g_+(x) - \alpha ^y g_-(y)}{\alpha ^{x+j}+\alpha ^y}\right) \right|\\&= \frac{1}{\left|\alpha ^{x+j}+\alpha ^y\right|^2} \left|{{\,\textrm{Im}\,}}\left( \alpha ^{2({{\,\textrm{Re}\,}}x + j)}g_+(x) - \alpha ^{\bar{x}+j+y}g_-(y) + \alpha ^{x+j+\bar{y}}g_+(x) - \alpha ^{2{{\,\textrm{Re}\,}}y}g_-(y)\right) \right|. \end{aligned}$$

Thus, using (4.61), noting that $|{{\,\textrm{Im}\,}}(\alpha ^{it}z)| \le |{{\,\textrm{Im}\,}}(z)| + |\sin (t\log \alpha )||z| \le |{{\,\textrm{Im}\,}}(z)| + |t\log \alpha ||z|$, for $t\in \mathbb {R}$ and $z\in \mathbb {C}$, and using (4.62) again to obtain the last two inequalities, we see that for $x,y\in \Xi _c$, where $r = \alpha ^{{{\,\textrm{Re}\,}}x+j}$ and $s= \alpha ^{{{\,\textrm{Re}\,}}y}$,

$$\begin{aligned} \left|{{\,\textrm{Im}\,}}{\tilde{q}}^+_j(x,y)\right|&\le \frac{1}{\alpha ^2(r+s)^2}\Big (r^2|{{\,\textrm{Im}\,}}g_+(x)| +s^2|{{\,\textrm{Im}\,}}g_-(y)| \\&\quad + rs\big (|{{\,\textrm{Im}\,}}g_-(y)| + |{{\,\textrm{Im}\,}}g_+(x)| + c|\log \alpha |(|g_-(y)|+|g_+(x)|)\big )\Big )\\&\le \frac{r|{{\,\textrm{Im}\,}}g_+(x)| +s|{{\,\textrm{Im}\,}}g_-(y)| + c|\log \alpha |\max \{s|g_-(y)|,r|g_+(x)|\}}{\alpha ^2(r+s)}\\&\le \alpha ^{-2}\left( \max \{|{{\,\textrm{Im}\,}}g_+(x)|,|{{\,\textrm{Im}\,}}g_-(y)|\} + c|\log \alpha |\max \{|g_-(y)|,|g_+(x)|\}\right) . \end{aligned}$$

Thus

$$\begin{aligned} \sup \limits _{x\in \Xi _c, y\in [0,1]}\left|{{\,\textrm{Im}\,}}{\tilde{q}}^+_j(x,y)\right|&\le \alpha ^{-2}\left( \Vert {{\,\textrm{Im}\,}}g_+\Vert _c + \alpha c \mathfrak {K}_{c,+}\right) \quad \text{ and }\\ \sup \limits _{x\in [0,1], y\in \Xi _c}\left|{{\,\textrm{Im}\,}}{\tilde{q}}^+_j(x,y)\right|&\le \alpha ^{-2}\left( \Vert {{\,\textrm{Im}\,}}g_-\Vert _c + \alpha c \mathfrak {K}_{c,-}\right) \end{aligned}$$

so that

$$\begin{aligned} \inf \limits _{x \in \Xi _c,y \in [0,1]}\left| 1 + {\tilde{q}}^+_j(x,y)^2\right|&\ge 1 - \sup \limits _{x \in \Xi _c,y \in [0,1]}\left|{{\,\textrm{Im}\,}}{\tilde{q}}^+_j(x,y)\right|^2\\&\ge 1 - \alpha ^{-4}\left( \Vert {{\,\textrm{Im}\,}}g_+\Vert _c + \alpha c \mathfrak {K}_{c,+}\right) ^2 \quad \text{ and }\\ \inf \limits _{x \in [0,1],y \in \Xi _c}\left| 1 + {\tilde{q}}^+_j(x,y)^2\right|&\ge 1 - \alpha ^{-4}\left( \Vert {{\,\textrm{Im}\,}}g_-\Vert _c + \alpha c \mathfrak {K}_{c,-}\right) ^2. \end{aligned}$$

Combining these estimates, and using (4.28) with f replaced by $f_-$, we get

$$\begin{aligned} \sup \limits _{x \in \Xi _c,y \in [0,1]} \left|T_j(x,y)\right|&\le \frac{|\log \alpha |\mathfrak {F}_{0,-}+\mathfrak {K}_{c,+}}{1 - \alpha ^{-4}\left( \left\Vert {{\,\textrm{Im}\,}}g_+\right\Vert _c + \alpha c \mathfrak {K}_{c,+}\right) ^2} \left( 1 + \mathfrak {F}_{c,+}^2\right) ^{1/4} \frac{\alpha ^{j/2}}{\alpha ^{j+2}+\alpha ^2} \quad \text{ and }\\ \sup \limits _{x \in [0,1],y \in \Xi _c} \left|T_j(x,y)\right|&\le \frac{|\log \alpha |\mathfrak {F}_{c,-}+\mathfrak {K}_{c,-}}{1 - \alpha ^{-4}\left( \left\Vert {{\,\textrm{Im}\,}}g_-\right\Vert _c + \alpha c \mathfrak {K}_{c,-}\right) ^2} \left( \frac{1 + \mathfrak {F}_{0,+}^2}{1-\mathfrak {I}_{c,-}^2}\right) ^{1/4} \frac{\alpha ^{j/2}}{\alpha ^{j+2}+\alpha ^2}. \end{aligned}$$

The estimates (4.55) and (4.56) and the other results for $\tilde{L}_t^+$ follow. $\square$

Let ${\tilde{B}}_{t,N}^\pm$ and ${\tilde{C}}^\pm _{t,N}$ be the $N\times N$ matrices defined by the right hand side of (4.31) with ${\tilde{K}}_t(\cdot ,\cdot )$ replaced by ${\tilde{K}}_t^\pm (\cdot ,\cdot )$ and ${\tilde{L}}_t^\pm (\cdot ,\cdot )$, respectively, and define the $2N\times 2N$ matrix $A_{t,N}$ by

$$\begin{aligned} A_{t,N}:= \begin{pmatrix} B_{t,N}^{-} &{} C_{t,N}^{-} \\ C_{t,N}^{+} &{} B_{t,N}^{+} \end{pmatrix}, \quad t\in \mathbb {R}, \;\; N\in \mathbb {N}. \end{aligned}$$

Recalling Remark 3.11, the matrix $A_{t,N}$ and the operator $K_{t,N}$ are related by Lemma 3.7(ii) and (iii). As in Sect. 4.2, our goal is to estimate the spectrum and spectral radius of $K_{t,N}$ via computing the spectra of approximations to $A_{t,N}$ for some fixed N and finitely many $t\in [-\pi ,\pi ]$.

To define these approximations, proceeding analogously to Sect. 4.2, for $M \in \mathbb {N}$ let $\tilde{K}_t^{\pm ,M}(\cdot ,\cdot )$ and $\tilde{L}_t^{\pm ,M}(\cdot ,\cdot )$ be approximations to $\tilde{K}_t^{\pm }(\cdot ,\cdot )$ and $\tilde{L}_t^{\pm }(\cdot ,\cdot )$, respectively, given by (4.47) and (4.48) but with the infinite series replaced by finite sums from $j = -M$ to M [cf. (4.32)]. Let $B_{t,N}^{\pm ,M}$ and $C_{t,N}^{\pm ,M}$ be the corresponding approximations to the matrices $B_{t,N}^{\pm }$ and $C_{t,N}^{\pm }$, so that

$$\begin{aligned} B_{t,N}^{\pm ,M}(p,q)= & {} \frac{1}{N}\tilde{K}_t^{\pm ,M}(x_{p,N},x_{q,N}),\\ C_{t,N}^{\pm ,M}(p,q)= & {} \frac{1}{N}\tilde{L}_t^{\pm ,M}(x_{p,N},x_{q,N}), \quad p,q=1,\ldots ,N, \end{aligned}$$

and let

$$\begin{aligned} A_{t,N}^M := \begin{pmatrix} B_{t,N}^{-,M} &{} C_{t,N}^{-,M} \\ C_{t,N}^{+,M} &{} B_{t,N}^{+,M} \end{pmatrix}, \quad t\in \mathbb {R}, \;\; M,N\in \mathbb {N}. \end{aligned}$$

(4.63)

Similarly, define the operators ${\tilde{K}}^{\pm ,M}_{t,N}$ and ${\tilde{L}}^{\pm ,M}_{t,N}$ as we defined ${\tilde{K}}^\pm _{t,N}$ and ${\tilde{L}}^\pm _{t,N}$ above Proposition 4.16, but replacing the kernels $\tilde{K}_t^{\pm }(\cdot ,\cdot )$ and $\tilde{L}_t^{\pm }(\cdot ,\cdot )$ in their definitions by the respective approximations $\tilde{K}_t^{\pm ,M}(\cdot ,\cdot )$ and $\tilde{L}_t^{\pm ,M}(\cdot ,\cdot )$, and let

$$\begin{aligned} {\tilde{K}}_{t,N}^M = \begin{pmatrix} {\tilde{K}}^{-,M}_{t,N} &{} {\tilde{L}}^{-,M}_{t,N}\\ {\tilde{L}}^{+,M}_{t,N} &{} {\tilde{K}}^{+,M}_{t,N} \end{pmatrix}, \quad t\in \mathbb {R},\;\; M,N\in \mathbb {N}. \end{aligned}$$

(4.64)

Then, similarly to (4.33), we have that, for $t\in \mathbb {R}$ and $M,N\in \mathbb {N}$ with $M \ge 2$,

$$\begin{aligned} \Vert A_{t,N}-A_{t,N}^M\Vert _\infty \le \Vert {\tilde{K}}_{t,N}-{\tilde{K}}_{t,N}^M\Vert _\infty \le C_1(M), \end{aligned}$$

(4.65)

where, using the notations of Remarks 4.9 and 4.18 and Proposition 4.17, and setting $\mathfrak {K}_0:= \mathfrak {K}_{0,+}=\mathfrak {K}_{0,-}$,

$$\begin{aligned} C_1(M)&:=\frac{1}{\pi }\max \Big \{\left( 1+\mathfrak {F}_{0,+}^2\right) ^{1/4}\big (2|\log \alpha |\mathfrak {F}_{0,+}\mathcal {B}_M(\alpha ) + (|\log \alpha |\mathfrak {F}_{0,-}+\mathfrak {K}_0)\mathcal {C}_M(\alpha )\big ),\nonumber \\&\quad \left( 1+\mathfrak {F}_{0,-}^2\right) ^{1/4}\big (2|\log \alpha |\mathfrak {F}_{0,-} \mathcal {B}_M(\alpha ) + (|\log \alpha |\mathfrak {F}_{0,+}+\mathfrak {K}_0)\mathcal {C}_M(\alpha )\big )\Big \}. \end{aligned}$$

(4.66)

Note that by (4.26) and (4.57), $C_1(M) = O(\alpha ^{M/2})$ as $M\rightarrow \infty$.

Arguing as in (4.35), we have also that, for $M,N\in \mathbb {N}$, $t\in \mathbb {R}$, $p,q=1,\ldots ,N$,

$$\begin{aligned} \left| \frac{\partial }{\partial t} B_{t,N}^{\pm ,M}(p,q)\right| \le B_N^{\pm ,M}(p,q) \quad \text{ and } \quad \left| \frac{\partial }{\partial t} C_{t,N}^{\pm ,M}(p,q)\right| \le C_N^{\pm ,M}(p,q), \end{aligned}$$

(4.67)

where $B_N^{\pm ,M}(p,q)$ is defined by (4.35) but with $p_j$, $q_j$, and f replaced by $p_j^\pm$, $q_j^\pm$ and $f_\pm$, respectively [compare (4.15) and (4.47)]. Similarly, $C_N^{\pm ,M}(p,q)$ is defined by the right hand side of (4.35) with $p_j$ and $q_j$ replaced by ${\tilde{p}}_j^\pm$ and ${\tilde{q}}_j^\pm$, respectively, and with f replaced by $f_\pm$ in the numerator, by $f_\mp$ in the denominator [compare (4.15) and (4.48)]. Thus, where

$$\begin{aligned} B_{N}^M := \begin{pmatrix} B_{N}^{-,M} &{} C_{N}^{-,M} \\ C_{N}^{+,M} &{} B_{N}^{+,M} \end{pmatrix}, \quad M,N\in \mathbb {N}, \end{aligned}$$

(4.68)

Eq. (4.36) holds (with the above definitions of $A_{t,N}$, $A_{t,N}^M$ and $B_N^M$) for all $s,t\in \mathbb {R}$, $M,N\in \mathbb {N}$. Note also, arguing as in (4.37) and (4.38), that $\Vert B_N^M\Vert _\infty$ is bounded uniformly for $M,N\in \mathbb {N}$, and that $\Vert {\tilde{K}}_{t,N}-{\tilde{K}}_{s,N}\Vert _\infty = O(|s-t|)$ as $|s-t|\rightarrow 0$, uniformly for $s,t\in \mathbb {R}$, $N\in \mathbb {N}$. Further, similarly to (4.39) and (4.66), $\Vert {\tilde{K}}_{t,N}\Vert _\infty \le C_3$ for $t\in \mathbb {R}$ and $N\in \mathbb {N}$, where¹⁸

$$\begin{aligned} C_3&:= \frac{1}{2\pi }\max \Big \{\left( 1+\mathfrak {F}_{0,+}^2\right) ^{1/4}\big (\mathfrak {G}_{0,+} \mathfrak {R}(\alpha )+4|\log \alpha |\mathfrak {F}_{0,+}\mathcal {B}^*_{10}(\alpha ) + (|\log \alpha |\mathfrak {F}_{0,-}+\mathfrak {K}_0)\mathcal {C}^*_{10}(\alpha )\big ), \nonumber \\&\quad \left( 1+\mathfrak {F}_{0,-}^2\right) ^{1/4}\big (\mathfrak {G}_{0,-} \mathfrak {R}(\alpha )+4|\log \alpha |\mathfrak {F}_{0,-}\mathcal {B}^*_{10}(\alpha ) + (|\log \alpha |\mathfrak {F}_{0,+}+\mathfrak {K}_0)\mathcal {C}^*_{10}(\alpha )\big )\Big \}, \end{aligned}$$

(4.69)

$\mathfrak {R}(\alpha ):=(1 + \alpha ^{1/2} + \alpha ^{-1/2})/(2\alpha ^2)$, $\mathcal {B}^*_{10}$ and $\mathcal {C}^*_{10}$ are defined by (4.25) and (4.58), and the other notations are defined in Proposition 4.17. Provided c is such that the conditions of Proposition 4.17 hold, we have also [cf. (4.40)], by Propositions 4.16 and 4.17, that (4.40) holds with

$$\begin{aligned} C_4 := 2e^{2\pi c}\widetilde{C}^*, \end{aligned}$$

(4.70)

where $\widetilde{C}^*$ is defined by the right hand side of (4.51), but with the norms on that right hand side replaced by the upper bounds in Proposition 4.17. Additionally, we replace the infinite sums in these upper bounds by the bounds $\mathcal {B}_{10}^*(\alpha )$ and $\mathcal {C}_{10}^*(\alpha )$.

The following result, which holds for every $\Gamma$ that satisfies our standing assumption (4.44), is identical to Theorem 4.13, except that ${\tilde{K}}_{t,N}^{M_N}$ and $A_{t,N}^{M_N}$ are defined here by (4.64) and (4.63), respectively. In this theorem $\sigma (\tilde{K}^M_{t,N})$ denotes the spectrum of ${\tilde{K}}^M_{t,N}$ either on $(L^2(0,1))^2$ or on $(C[0,1])^2$ (cf. (4.19)), which coincides with $\{0\}\cup \sigma (A^M_{t,N})$ by Lemma 3.7(ii) and Remark 3.11, and $\sigma (D_\Gamma )$ denotes the spectrum of $D_\Gamma$ on $L^2(\Gamma )$, which coincides with the essential spectrum by Corollary 4.2. The proof of this result follows that of Theorem 4.13, noting that the argument of Lemma 4.12 applies to each of the operator families ${\tilde{K}}_{t,N}^{\pm ,M}$ and ${\tilde{L}}_{t,N}^{\pm ,M}$ so that [where ${\tilde{K}}_{t,N}^{M}$ is defined by (4.64)] $\{{\tilde{K}}_{t,N}^M:t\in [-\pi ,\pi ],M,N\in \mathbb {N}\}$ is collectively compact.

Theorem 4.19

Choose sequences $(m_N)_{N\in \mathbb {N}}, (M_N)_{N\in \mathbb {N}}\subset \mathbb {N}$ such that $m_N, M_N\rightarrow \infty$ as $N\rightarrow \infty$, and for each N, let $T_N:=\{\pm (k-1/2)\pi /m_N:k=1,\ldots ,m_N\}$. Then, as $N\rightarrow \infty$,

(4.71)

Remark 4.20

Using Remark 3.11, we see that Remark 4.14 applies also to $\sigma ^N(D_\Gamma )$ given by (4.71).

Our second main result of this subsection, obtained by applying Theorem 3.8, noting Remark 3.11, is proved in the same way as the analogous result, Theorem 4.15, in the one-sided case. In this theorem $A_{t,N}^M$, $B_N^M$, $C_1(M)$, $C_3$, and $C_4$, are defined by (4.63), (4.68), (4.66), (4.69), and (4.70), respectively.

Theorem 4.21

Under our standing assumption (4.44), define $g_\pm$ by (4.45) so that, by Lemma 4.7, $g_\pm$ have analytic continuations to $\Sigma _c$, for some $c>0$, such that $g_\pm$, $g_\pm ^\prime$, and $g_\pm ^{\prime \prime }$ are bounded on $\Sigma _c$, and, without loss of generality, assume that (4.54) holds. Suppose that $\rho _0>0$, $m,M,N\in \mathbb {N}$, with $M\ge 2$. For $k=1,\ldots ,m$, set $t_k:= (k-1/2)\pi /m$, suppose that $\rho (A^M_{t_k,N})<\rho _0$ for $k=1,\ldots ,m$, and recursively define $\mu _{k,\ell }$, for $\ell =1,2,\ldots$, by $\mu _{k,1}:= \rho _0$, and by (4.42). Further, for $k=1,\ldots ,m$, let $n_k$ denote the smallest integer such that $\sum _{\ell =1}^{n_k} \nu _{k,\ell } \ge 4\pi \rho _0$, and let

$$\begin{aligned} R_{m,M,N}:=\min _{{\mathop {\ell =1,\ldots ,n_k}\limits ^{k=1,\ldots ,m}}}\left( \frac{\nu _{k,\ell }}{4}+\frac{\nu _{k,\ell +1}}{2}\right) . \end{aligned}$$

If

$$\begin{aligned} \mathcal {L}_c(f,N):= & {} \frac{C_4}{e^{2\pi Nc}-1} < \mathcal {R}_c(f,m,M,N)\nonumber \\:= & {} r^2_0\left( 1 + C_3\left( R_{m,M,N} - C_1(M)- \frac{\pi }{2m}\Vert B_N^M\Vert _\infty \right) ^{-1} \right) ^{-1}, \end{aligned}$$

(4.72)

then $\rho (D_\Gamma ;L^2(\Gamma )) < \rho _0$. Conversely, if $\rho (D_\Gamma ;L^2(\Gamma )) < \rho _0$, then, provided m and M are sufficiently large, there exists $N_0\in \mathbb {N}$ such that (4.72) holds and $\rho (A_{t,N}^M)<\rho _0$ for $t\in [0,\pi ]$ and all $N\ge N_0$.

Remark 4.22

The condition (4.72) can be written as

$$\begin{aligned} S_c(\Gamma ,m,M,N)<0, \end{aligned}$$

where $S_c(\Gamma ,m,M,N):= \mathcal {L}_c(f,N)-\mathcal {R}_c(f,m,M,N)$. For any $c>0$ such that the conditions of the first sentence of the above theorem are satisfied (and these conditions are necessarily satisfied for all sufficiently small $c>0$), we note that:

• (i) The proof of the above theorem, which follows that of Theorem 4.15, shows that, provided $\rho (D_\Gamma ;L^2(\Gamma )) < \rho _0$, $R_{m,M,N}$ is positive and bounded away from zero for all sufficiently large $m,M,N\in \mathbb {N}$, so that the same holds for $\mathcal {R}_c(f,m,M,N)$. Since also $\mathcal {L}_c(f,N)\rightarrow 0$ as $N\rightarrow \infty$, $S_c(\Gamma ,m,M,N)<0$ for all sufficiently large m, M, and N, if $\rho (D_\Gamma ;L^2(\Gamma )) < \rho _0$.

• (ii) For fixed $m,M,N\in \mathbb {N}$, evaluation of the functionals $\mathcal {L}(f,N)$ and $\mathcal {R}_c(f,m,M,N)$ requires inputs relating to the functions $f_\pm$ defined by (4.45), namely: the constant $\alpha \in (0,1)$ and the bounds $\Vert g_\pm \Vert _0$, $\Vert g_\pm ^\prime \Vert _0$, and $\Vert g_\pm ^{\prime \prime }\Vert _0$ on g defined by (4.45) (to evaluate $C_1(M)$ and $C_3$); the bounds $\Vert g_\pm \Vert _c$, $\Vert g_\pm ^\prime \Vert _c$, and $\Vert g_\pm ^{\prime \prime }\Vert _c$ (to evaluate $C_4$); the values $f_\pm (\alpha ^{x_{p,N}+j})$, $f_\pm ^\prime (\alpha ^{x_{p,N}})$, and $f_\pm ^{\prime \prime }(\alpha ^{x_{p,N}})$, for $p=1,\ldots , N$ and $j=-M,\ldots ,M$ (to compute the entries of the matrices $A_{t_k,N}^M$, for $k=1,\ldots ,m$).

• (iii) If exact values of $\Vert g_\pm \Vert _0$, $\Vert g_\pm ^\prime \Vert _0$, $\Vert g_\pm ^{\prime \prime }\Vert _0$, $\Vert g_\pm \Vert _c$, $\Vert g_\pm ^\prime \Vert _c$, and $\Vert g_\pm ^{\prime \prime }\Vert _c$ are not available, it is enough to replace them with upper bounds which leads to upper bounds $\widehat{C}_1(M)$, $\widehat{C}_3$ and $\widehat{C}_4$ for $C_1(M)$, $C_3$ and $C_4$, respectively (with $\widehat{C}_1(M)\rightarrow 0$ as $M\rightarrow \infty$ at the same rate as $C_1(M)$). Let $\widehat{\mathcal {L}}_c(f,N)$ and $\widehat{\mathcal {R}}_c(f,m,M,N)$ be defined by (4.43) with $C_1(M)$, $C_3$, $C_4$, replaced by their upper bounds, so that $\mathcal {L}_c(f,N)\le \widehat{\mathcal {L}}_c(f,N)$ and $\widehat{\mathcal {R}}_c(f,m,M,N)\le {\mathcal {R}}_c(f,m,M,N)$. Then the conclusions of Theorem 4.21 hold with $\mathcal {L}_c(f,N)$ and $\mathcal {R}_c(f,m,M,N)$ replaced by $\widehat{\mathcal {L}}_c(f,N)$ and $\widehat{\mathcal {R}}_c(f,m,M,N)$, respectively; in particular $S_c(\Gamma ,m,M,N)<0$ if $\widehat{\mathcal {L}}_c(f,N) < \widehat{\mathcal {R}}_c(f,m,M,N)$ and, provided $\rho (D_\Gamma ;L^2(\Gamma )) < \rho _0$, $\widehat{\mathcal {L}}_c(f,N) < \widehat{\mathcal {R}}_c(f,m,M,N)$ for all sufficiently large m, M, and N.

4.4 The 2D case: lower bounds for the numerical range

This paper is motivated by the question at the end of Sect. 1.1 which notes that there exist (e.g., [11, Fig. 3]) 2D examples of $\Gamma \in \mathcal {D}$ with $w_{{{\,\textrm{ess}\,}}}(D_\Gamma )\gg \frac{1}{2}$ and asks: is $\rho _{L^2(\Gamma ),{{\,\textrm{ess}\,}}}\ge \frac{1}{2}$ for any of these examples? We will explore this in Sect. 6 where we will see that $\rho _{L^2(\Gamma ),{{\,\textrm{ess}\,}})}< \frac{1}{2}$ for each example we treat, supporting Conjecture 1.2. We will also see that $w_{{{\,\textrm{ess}\,}}}(D_\Gamma )>\frac{1}{2}$ for at least some of these examples, indeed that $W_{{{\,\textrm{ess}\,}}}(D_\Gamma )\supset B_R:=\{z\in \mathbb {C}:|z|<R\}$ for R substantially larger than $\frac{1}{2}$.

As the route to obtain these estimates for $W_{{{\,\textrm{ess}\,}}}(D_\Gamma )$, in this subsection we obtain lower bounds¹⁹ for $W(D_\Gamma )=W_{{{\,\textrm{ess}\,}}}(D_\Gamma )$ in the case when $\Gamma$ is a dilation invariant graph, precisely in the 2D case when either (4.8) or (4.44) applies. It is enough to restrict attention to the one-sided case (4.8) as one easily sees (e.g., [11, Sect. 2.3]) that if $\Gamma$ satisfies (4.44) and $\widetilde{\Gamma }:= \{(x,f(x)):x>0\}$, then $\widetilde{\Gamma }$ satisfies (4.8) and

$$\begin{aligned} W(D_{\widetilde{\Gamma }})\subset W(D_\Gamma ). \end{aligned}$$

(4.73)

We obtain lower bounds for $W(D_\Gamma )$ in the case that (4.8) applies via the Nyström method that we used in Sect. 4.2 to approximate $\sigma (D_\Gamma )$. Firstly, by Corollary 4.4, and since, for $t\in [-\pi ,\pi ]$, ${\tilde{L}}_t$, defined in the proof of Proposition 4.10, is unitarily equivalent to ${\tilde{K}}_{t}$ defined by (4.13), which is unitarily equivalent to $K_{t}$, we have that $W({\tilde{L}}_t)\subset W(D_\Gamma )$. For $p\in \mathbb {N}_{0}$, let $\mathcal {T}_p\subset L^2(0,1)$ denote the set of trigonometric polynomials of degree at most p. An orthonormal basis of $\mathcal {T}_p$ is given by $\left\{ e_{-p},\ldots ,e_p\right\}$, where $e_j(x):= e^{2\pi ijx}$. Consider the restricted numerical range

$$\begin{aligned} W_p({\tilde{L}}_t):= \left\{ \langle {\tilde{L}}_t\phi ,\phi \rangle : \phi \in \mathcal {T}_p,\left\Vert \phi \right\Vert _{L^2(0,1)} = 1\right\} . \end{aligned}$$

It is clear that $W_p({\tilde{L}}_t) \subset W({\tilde{L}}_t)$ for $t\in [-\pi ,\pi ]$, and that (e.g., [11, Sect. 2.3]) $W_p({\tilde{L}}_t)=W(T^{p,t})$, where $T^{p,t} = (T^{p,t}_{jk})_{j,k = -p}^p$ is the $(2p+1) \times (2p+1)$ matrix with $T^{p,t}_{jk}:= \langle {\tilde{L}}_t e_k,e_j\rangle$, $j,k=-p,\dots ,p$. Recalling from the proof of Proposition 4.10 that ${\tilde{L}}_t$ has kernel $\tilde{L}_t(x,y) = e^{i(x-y)t}\tilde{K}_t(x,y)$, we will approximate $T^{p,t}$ by $T^{p,t,N}$, where

$$\begin{aligned} T^{p,t,N}_{jk} := \frac{1}{N^2}\sum \limits _{m,n = 1}^N e^{i(x_{m,N}-x_{n,N})t}{\tilde{K}}_t(x_{m,N},x_{n,N})e_k(x_{n,N})\overline{e_j(x_{m,N})}, \quad j,k=-p,\ldots ,p, \end{aligned}$$

(4.74)

recalling that $x_{m,N}:= \tfrac{1}{N}\left( m-\tfrac{1}{2}\right)$, $m=1,\ldots ,N$, and then approximate $T^{p,t,N}$ further by $T^{p,t,N,M}$, defined by (4.74) but with ${\tilde{K}}_{t}$, given by the infinite sum (4.15), replaced by ${\tilde{K}}_{t}^M$, given by the finite sum (4.32). The following result, which uses the notation (4.34), will enable us to estimate the difference between $W_p({\tilde{L}}_t)=W(T^{p,t})$ and $W(T^{p,t,N,M})$.

Proposition 4.23

Let f and $c>0$ be as in Proposition  4.8, in particular satisfying (4.21), so that ${\tilde{L}}_t(x,\cdot )$ and ${\tilde{L}}_t(\cdot ,y)$ are analytic and bounded in $\Sigma _c$ for all $x,y,t\in \mathbb {R}$, let ${\tilde{L}}^M_t(x,y):= e^{i(x-y)}{\tilde{K}}^M(x,y)$, for $x,y\in \mathbb {R}$ and $M\in \mathbb {N}$, and suppose that g is a bounded, analytic, 1-periodic function on $\Sigma _c$. Then, for $M,N\in \mathbb {N}$ and $t\in [-\pi ,\pi ]$,

$$\begin{aligned}{} & {} \left|\langle {\tilde{L}}_t g,g\rangle - \frac{1}{N^2}\sum \limits _{m,n = 1}^N \tilde{L}^{M}_{t}(x_{m,N},x_{n,N})g(x_{n,N})\overline{g(x_{m,N})}\right|\\{} & {} \quad \le 2\left\Vert g\right\Vert _0\left\Vert g\right\Vert _c e^{\pi c}\frac{\Vert {\tilde{K}}_t\Vert _{0,c} + \Vert {\tilde{K}}_t\Vert _{c,0}}{e^{2\pi Nc}-1} + C_1(M). \end{aligned}$$

Proof

$$\begin{aligned}&\left|\langle {\tilde{L}}_t g,g\rangle - \frac{1}{N^2}\sum \limits _{m,n = 1}^N {\tilde{L}}_t(x_{m,N},x_{n,N}) g(x_{n,N})\overline{g(x_{m,N})}\right|\\&\quad \le \int _0^1 \left|g(x)\right|\left|\int _0^1 {\tilde{L}}_t (x,y)g(y) \, \textrm{d}y - \frac{1}{N}\sum \limits _{n = 1}^N {\tilde{L}}_t (x,x_{n,N})g(x_{n,N})\right| \, \textrm{d}x\\&\qquad +\, \frac{1}{N} \sum \limits _{n = 1}^N \left|\int _0^1 \overline{g(x)}{\tilde{L}}_t (x,x_{n,N}) \, \textrm{d}x - \frac{1}{N}\sum \limits _{m = 1}^N \overline{g(x_{m,N})}{\tilde{L}}_t (x_{m,N},x_{n,N})\right|\left|g(x_{n,N})\right|\\&\quad \le \left\Vert g\right\Vert _0\frac{2\Vert {\tilde{L}}_t \Vert _{0,c}\left\Vert g\right\Vert _c}{e^{2\pi Nc}-1} + \frac{2\left\Vert g\right\Vert _c\Vert {\tilde{L}}_t \Vert _{c,0}}{e^{2\pi Nc}-1}\left\Vert g\right\Vert _0, \end{aligned}$$

by Theorem 4.11, noting that g and, from the proof of Proposition 4.10, ${\tilde{L}}_t (x,\cdot )$, for $x\in \mathbb {R}$, and ${\tilde{L}}_t (\cdot ,y)$, for $y\in \mathbb {R}$, are all 1-periodic. The claimed result now follows immediately from (4.33) and by noting that, from the proof of Proposition 4.10, $\Vert {\tilde{L}}_t\Vert _{0,c}\le e^{\pi c} \Vert {\tilde{K}}_t\Vert _{0,c}$ and $\Vert {\tilde{L}}_t\Vert _{c,0}\le e^{\pi c} \Vert {\tilde{K}}_t\Vert _{c,0}$. $\square$

Clearly, $g \in \mathcal {T}^p$ with $\left\Vert g\right\Vert _{L^2(0,1)} = 1$ if and only if

$$\begin{aligned} g = \sum \limits _{j = -p}^p c_je_j \quad \text {with} \quad \sum \limits _{j = -p}^p \left|c_j\right|^2 = 1, \end{aligned}$$

(4.75)

in which case

$$\begin{aligned} \left\Vert g\right\Vert _c \le \sum \limits _{j = -p}^p \left|c_j\right|\left\Vert e_j\right\Vert _c \le \sum \limits _{j = -p}^p \left|c_j\right|\left\Vert e_j\right\Vert _c \le \sum \limits _{j = -p}^p \left|c_j\right|e^{2\pi p c} \le \sqrt{2p+1} \; e^{2\pi p c}. \end{aligned}$$

The following corollary follows immediately from this observation and Proposition 4.23, noting that, if g and $c_{-p},\ldots ,c_p$ are related by (4.75), then the summation in Proposition 4.23 coincides with $\sum _{j,k=-p}^p T_{j,k}^{p,t,N,M}c_k\overline{c_j}\in W(T^{p,t,N,M})$.

Corollary 4.24

For $p\in \mathbb {N}_0$, $M,N\in \mathbb {N}$, $t\in [-\pi ,\pi ]$, where $c>0$ is as in Proposition 4.23,

$$\begin{aligned}{} & {} d_H(W_p({\tilde{L}}_t ),W(T^{p,t,N,M})) \le C_7(p,N,M) \\{} & {} \quad := 2(2p+1)e^{\pi c(2p+1)} \frac{\Vert {\tilde{K}}_{t}\Vert _{0,c} + \Vert {\tilde{K}}_{t}\Vert _{c,0}}{e^{2\pi Nc}-1} + C_1(M). \end{aligned}$$

We can approximate $W(T^{p,t,N,M})$ by a standard method that dates back to Johnson [31]. Choose $n\in \mathbb {N}$ and, for $\ell =0,\ldots ,n$, let $\theta _\ell := 2\pi \ell /n$, let $\lambda _\ell$ denote the largest eigenvalue of ${{\,\textrm{Re}\,}}(e^{-i\theta _\ell }T^{p,t,N,M})$ and $x_\ell$ an associated unit eigenvector, and let $z_\ell := \langle T^{p,t,N,M}x_\ell ,x_\ell \rangle$, for $\ell =0,1,\ldots ,n-1$, and $z_n:= z_0$. Then $W_n:= \mathop {\textrm{conv}}\limits (\{z_0,\ldots ,z_n\})\subset W(T^{p,t,N,M})$ and (Johnson [31]) as $n\rightarrow \infty$. The following simple corollary of the above results, in which we use the notations just introduced, will enable us to show that $B_R \subset W({\tilde{L}}_t )\subset W(D_\Gamma )$ for concrete values of $R>0$ in the examples we treat in Sect. 6.

Corollary 4.25

Suppose that $p\in \mathbb {N}_0$, $M,N,n\in \mathbb {N}$, $t\in [-\pi ,\pi ]$, that $c>0$ is as in Proposition 4.23, and that 0 is an interior point of $W_n$, in which case $z_\ell = \exp (i\gamma _\ell )|z_\ell |$, $\ell =0,\ldots ,n$, with $\gamma _0\le \gamma _1\le \ldots \le \gamma _n=\gamma _0+2\pi$. Then, where $\theta _{\max }:= \max _{\ell =1,\ldots ,n} (\gamma _\ell -\gamma _{\ell -1})\in (0,2\pi ]$ and $R_{\min }:= \min _{\ell =0,\ldots ,n}|z_\ell |>0$, if $R^*:= R_{\min }\cos (\theta _{\max }/2)-C_7(p,t,N,M) > 0$, then $\overline{B_{R^*}}\subset W(D_\Gamma )$.

Proof

Under the above assumptions, $\overline{B_R}\subset W_n$, where $R:= R_{\min }\cos (\theta _{\max }/2)$. Since $W_n\subset W(T^{p,t,N,M})$ and $W_p({\tilde{L}}_t )\subset W({\tilde{L}}_t )\subset W(D_\Gamma )$, it follows that $\overline{B_{R^*}}\subset W(D_\Gamma )$ by Corollary 4.24. $\square$

5 Localization and deformation

Let $\Omega _- \subset {\mathbb {R}}^d$, $d \ge 2$, be a Lipschitz domain with boundary $\Gamma = \partial \Omega _-$ and outward-pointing unit normal vector $n_y = n^\Gamma _y$ at almost every $y \in \Gamma$. If $\psi :{\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$ is a $C^1$-diffeomorphism, then $\psi (\Omega _-)$ is a Lipschitz domain with boundary $\psi (\Gamma )$ by Hofmann et al. [29, Theorem 4.1]. Note that this is no longer true if $\psi$ is only assumed to be a bi-Lipschitz map (see [21, Lemma 1.2.1.4], and the discussion in [29]). As before, $D_\Gamma$ will be the double-layer operator on $\Gamma$. To simplify notation we will use $\sim$ to denote equality up to compact operators, and abbreviate $D_\Gamma -\lambda I$ as $D_\Gamma -\lambda$. For $x\in \mathbb {R}^d$ and $r>0$, $B(x,r):= \{y\in \mathbb {R}^d:|y-x|<r\}$.

5.1 Localization without deformation

To start, we will consider domains with locally-dilation-invariant boundaries $\Gamma \in \mathscr {D}$; recall Definition 1.3. Note that being locally dilation invariant at x is equivalent to the existence of an isometry $\psi _x: \mathbb {R}^d \rightarrow \mathbb {R}^d$ with $\psi _x(x) = 0$ and $\psi _x\big (B(x,\delta (x)) \cap \Gamma \big ) = B(0,\delta (x)) \cap \Gamma _x$, for some $\delta (x)>0$, where $\Gamma _x$ is the graph of a Lipschitz continuous function with $\alpha \Gamma _x = \Gamma _x$. We also assume that $\psi _x$ preserves the orientation of the outer normal vector field, that is, the outward-pointing normal vector field on $B(x,\delta (x)) \cap \Gamma$ is mapped to the upwards-pointing normal vector field on $B(0,\delta (x)) \cap \Gamma _x$. Similarly, $\Gamma$ is locally $C^1$ at x if there is a $\delta (x) > 0$ and an Euclidean map $\psi _x: \mathbb {R}^d \rightarrow \mathbb {R}^d$ with $\psi _x(x) = 0$ and $\psi _x(B(x,\delta (x)) \cap \Gamma ) = B(0,\delta (x)) \cap \Gamma _x$, where $\Gamma _x$ is the graph of a $C^1$-function with compact support. Again, we assume that $\psi _x$ preserves the orientation of the outer normal vector field. We will use extensively below the notation $\delta (x)$, for $x\in \Gamma$, which will have one of the above meanings, depending on whether $\Gamma$ is locally dilation invariant at x or is $C^1$ at x.

Note that $D_{\Gamma }$ is compact for every $C^1$ domain [19], as we recalled in Sect. 1.2; this implies that also $D_{\widetilde{\Gamma }}$ is compact if $\widetilde{\Gamma }$ is the graph of a $C^1$-function with compact support. Consequently, if $\Gamma$ consists of different parts that are separated by $C^1$ areas, we are able to localize using sharp cut-off functions. In the following we equip $\Gamma$ with the topology induced from $\mathbb {R}^d$ and the standard surface measure. In particular, $\chi _E$ denotes the characteristic function of a subset $E \subset \Gamma$ that is measurable with respect to the surface measure on $\Gamma$.

Lemma 5.1

Let $E \subset \Gamma$ be a measurable subset and assume that $\Gamma$ is locally $C^1$ at every $x \in \partial E$. Then $D_{\Gamma }$ essentially commutes with $\chi _E$, i.e., the commutator $[D_\Gamma ,\chi _E]:= D_\Gamma \chi _E-\chi _ED_\Gamma$ is compact.

Proof

Since $\partial E$ is a compact subset of $\Gamma$, there is an $\varepsilon > 0$ such that

$$\begin{aligned} (\partial E + B(0,\varepsilon )) \cap \Gamma \subset \bigcup \limits _{x \in \partial E} (B(x,\delta (x)) \cap \Gamma ). \end{aligned}$$

Set $E_o:= (\partial E + B(0,\tfrac{\varepsilon }{2})) \cap \Gamma$. We know that $\chi _AD_{\Gamma }\chi _B$ is compact, even Hilbert–Schmidt, whenever A and B have positive distance from each other. We have

$$\begin{aligned} E = (E \cap E_o) \cup (E \setminus E_o) \quad \text{ and } \quad E^c = (E^c \cap E_o) \cup (E^c \setminus E_o), \end{aligned}$$

where $E^c:= \Gamma \setminus E$. By inspection, we see that the four sets on the right of these equalities, if non-empty, have pairwise positive distance from each other except for $E \cap E_o$ and $E^c \cap E_o$. We thus have

$$\begin{aligned} \chi _ED_{\Gamma }\chi _{E^c}&\sim \chi _{E \cap E_o}D_{\Gamma }\chi _{E^c \cap E_o} = \chi _E\chi _{E_o}D_{\Gamma }\chi _{E_o}\chi _{E^c},\\ \chi _{E^c}D_{\Gamma }\chi _E&\sim \chi _{E^c \cap E_o}D_{\Gamma }\chi _{E \cap E_o} = \chi _{E^c}\chi _{E_o}D_{\Gamma }\chi _{E_o}\chi _E. \end{aligned}$$

The operator $\chi _{E_o}D_{\Gamma }\chi _{E_o}$ is compact because $\overline{E}_o$ is $C^1$ everywhere by construction. We conclude that $[D_{\Gamma },\chi _E] = \chi _{E^c}D_{\Gamma }\chi _E - \chi _ED_{\Gamma }\chi _{E^c}$ is compact as well. $\square$

In particular, this shows that if $\Gamma$ is locally $C^1$ at some point $x \in \Gamma$, then $D_\Gamma$ is not Fredholm, since if $E = B(x,\tfrac{\delta (x)}{2})$, then $\chi _E D_\Gamma \chi _E$ is compact, and Lemma 5.1 implies that $D_\Gamma \sim \chi _E D_\Gamma \chi _E + \chi _{E^c} D_\Gamma \chi _{E^c}$. Hence

$$\begin{aligned} 0\in \sigma _{{{\,\textrm{ess}\,}}}(\chi _E D_\Gamma \chi _E) \subset \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma }). \end{aligned}$$

(5.1)

More generally, if $E_1,\ldots ,E_N$ are measurable subsets of $\Gamma$ such that $\Gamma = \cup _{j = 1}^N E_j$, $E_j \cap E_k = \emptyset$ for $j \ne k$ and $\Gamma$ is locally $C^1$ at every $x \in \cup _{j = 1}^N \partial E_j$, then $D_\Gamma \sim \sum _{j=1}^N \chi _{E_j} D_\Gamma \chi _{E_j}$ and thus

$$\begin{aligned} \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma }) = \bigcup \limits _{j = 1}^N \sigma _{{{\,\textrm{ess}\,}}}(\chi _{E_j}D_{\Gamma }\chi _{E_j}). \end{aligned}$$

(5.2)

Equation (5.2) is a localisation result. We have also the following, more substantial localisation result (cf. [11, 15, 16, 48]) for the case where $\Gamma \in \mathscr {D}$.

Theorem 5.2

Let $\Omega _- \subset \mathbb {R}^d$ be a bounded Lipschitz domain such that $\Gamma = \partial \Omega _- \in \mathscr {D}$, and pick $x_1,\ldots ,x_N \in \Gamma$ for which $\Gamma \subset \bigcup \nolimits _{j = 1}^N B(x_j, \delta (x_j))$. Then

$$\begin{aligned} \sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma ) = \bigcup \limits _{x \in \Gamma } \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _x}) = \bigcup \limits _{j = 1}^N \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _{x_j}}). \end{aligned}$$

Proof

Let $x \in \Gamma$. If $\Gamma$ is locally $C^1$ at x, then, by (5.1) and since $D_{\Gamma _x}$ is compact, $\sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _x}) = \left\{ 0\right\} \subset \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma })$. So assume that $\Gamma$ is locally dilation invariant at x. Then there is an $\alpha \in (0,1)$ such that $\alpha \Gamma _x = \Gamma _x$. Let $\eta :\Gamma \rightarrow [0,1]$ be a Lipschitz continuous function with support in $B(x,\delta (x)) \cap \Gamma$ that is equal to 1 in a neighbourhood of x. Let $\psi _x :\mathbb {R}^d \rightarrow \mathbb {R}^d$ be an isometry with $\psi _x(x) = 0$ and $\psi _x(B(x,\delta (x)) \cap \Gamma ) = B(0,\delta (x)) \cap \Gamma _x$. Define $\Psi _x :L^2(\Gamma _x) \rightarrow L^2(\Gamma )$ by

$$\begin{aligned} \Psi _x\phi (y) = {\left\{ \begin{array}{ll} \phi (\psi _x(y)) &{} \text {if } y \in B(x,\delta (x)) \cap \Gamma , \\ 0 &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

which has adjoint $\Psi _x^\prime :L^2(\Gamma )\rightarrow L^2(\Gamma _x)$ given by

$$\begin{aligned} \Psi _x^\prime \phi (y) = {\left\{ \begin{array}{ll} \phi (\psi _x^{-1}(y)) &{} \text {if } y \in B(0,\delta (x)) \cap \Gamma _x, \\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

$\Psi _x$ and $\Psi _x^\prime$ are partial isometries with initial spaces $L^2\big (B(0,\delta (x)) \cap \Gamma _x\big )$ and $L^2\big (B(x,\delta (x)) \cap \Gamma \big )$, respectively. For $m\in \mathbb {N}$ the composition $\Psi _x^\prime \eta ^{m}\Psi _x:L^2(\Gamma _x)\rightarrow L^2(\Gamma _x)$ is the operation of multiplication by $(\eta ')^m$, where $\eta '(y):= \eta (\psi ^{-1}_x(y))$, for $y\in B(x,\delta (x)) \cap \Gamma _x$, $\eta '(y):= 0$, $y\in \Gamma _x{\setminus } B(x,\delta (x))$.

Let $V_{\alpha }$ be the unitary dilation as defined in (4.2) but with $\Gamma$ replaced by $\Gamma _x$. Then

$$\begin{aligned} V_{\alpha }^n\eta 'V_{\alpha }^{-n}\phi (y) = {\left\{ \begin{array}{ll} \eta (\psi _x^{-1}(\alpha ^n y))\phi (y) &{} \text {if } \alpha ^n y \in B(0,\delta (x)) \cap \Gamma _x, \\ 0 &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

for $n \in \mathbb {N}$, $y \in \Gamma _x$ and $\phi \in L^2(\Gamma _x)$. By dominated convergence, it follows that $V_{\alpha }^n\eta 'V_{\alpha }^{-n}\phi \rightarrow \phi$ as $n\rightarrow \infty$ for all $\phi \in L^2(\Gamma _x)$. (We can see, similarly, that, for every $m\in \mathbb {N}$, $V_{\alpha }^n(\eta ')^mV_{\alpha }^{-n}\phi \rightarrow \phi$ as $n\rightarrow \infty$ for all $\phi \in L^2(\Gamma _x)$.) Using that $\psi _x$ is an isometry, we further observe that $\Psi _x^\prime \eta D_{\Gamma }\eta \Psi _x = \eta 'D_{\Gamma _x}\eta '$. This implies, since also $D_{\Gamma _x}V_\alpha = V_\alpha D_{\Gamma _x}$, that

$$\begin{aligned} V_{\alpha }^n\Psi _x^\prime \eta D_{\Gamma }\eta \Psi _xV_{\alpha }^{-n} = V_{\alpha }^n \eta ' D_{\Gamma _x} \eta ' V_{\alpha }^{-n} = (V_{\alpha }^n \eta ' V_{\alpha }^{-n})D_{\Gamma _x}(V_{\alpha }^n \eta ' V_{\alpha }^{-n}) \rightarrow D_{\Gamma _x} \end{aligned}$$

(5.3)

in the strong operator topology.

Now assume that $D_{\Gamma } - \lambda$ is Fredholm and let $A :L^2(\Gamma ) \rightarrow L^2(\Gamma )$ be a Fredholm regularizer of $D_{\Gamma } - \lambda$, so that the products $A(D_{\Gamma } - \lambda )$ and $(D_{\Gamma } - \lambda )A$ are compact perturbations of the identity. Then, for any $\phi \in L^2(\Gamma _x)$,

$$\begin{aligned} \left\Vert \phi \right\Vert \le \left\Vert V_{\alpha }^n\Psi _x^\prime \eta A \eta (D_{\Gamma } - \lambda ) \eta \Psi _xV_{\alpha }^{-n}\phi \right\Vert + \left\Vert V_{\alpha }^n(I - \Psi _x^\prime \eta A \eta (D_{\Gamma } - \lambda ) \eta \Psi _x)V_{\alpha }^{-n}\phi \right\Vert . \end{aligned}$$

(5.4)

The first term can be estimated as

$$\begin{aligned} \left\Vert V_{\alpha }^n\Psi _x^\prime \eta A \eta (D_{\Gamma } - \lambda ) \eta \Psi _xV_{\alpha }^{-n}\phi \right\Vert&\le \left\Vert A\right\Vert \left\Vert \eta (D_{\Gamma } - \lambda ) \eta \Psi _xV_{\alpha }^{-n}\phi \right\Vert \\&= \left\Vert A\right\Vert \left\Vert V_{\alpha }^n\Psi _x^\prime \eta (D_{\Gamma } - \lambda ) \eta \Psi _xV_{\alpha }^{-n}\phi \right\Vert , \end{aligned}$$

where we used that $V_{\alpha }$ is an isometry, $\Psi _x^\prime$ is an isometry on the range of the multiplication operator $\eta$, and $\left\Vert \eta \right\Vert \le 1$. By (5.3),

$$\begin{aligned} \left\Vert V_{\alpha }^n\Psi _x^\prime \eta (D_{\Gamma } - \lambda ) \eta \Psi _xV_{\alpha }^{-n}\phi \right\Vert \rightarrow \left\Vert (D_{\Gamma _x} - \lambda )\phi \right\Vert , \end{aligned}$$

as $n\rightarrow \infty$. The second term in (5.4) can be estimated as

$$\begin{aligned}&\left\Vert V_{\alpha }^n(I - \Psi _x^\prime \eta A \eta (D_{\Gamma } - \lambda ) \eta \Psi _x)V_{\alpha }^{-n}\phi \right\Vert \\&\quad \le \left\Vert V_{\alpha }^n(I - \Psi _x^\prime \eta A(D_{\Gamma } - \lambda ) \eta ^2\Psi _x)V_{\alpha }^{-n}\phi \right\Vert + \left\Vert V_{\alpha }^n\Psi _x^\prime \eta A [\eta ,D_{\Gamma }] \eta \Psi _xV_{\alpha }^{-n}\phi \right\Vert \\&\quad \le \left\Vert V_{\alpha }^n(I - \eta '^3)V_{\alpha }^{-n}\phi \right\Vert + \left\Vert V_{\alpha }^n\Psi _x^\prime \eta (A(D_{\Gamma } - \lambda ) - I) \eta ^2\Psi _xV_{\alpha }^{-n}\phi \right\Vert \\&\qquad +\, \left\Vert V_{\alpha }^n\Psi _x^\prime \eta A [\eta ,D_{\Gamma }] \eta \Psi _xV_{\alpha }^{-n}\phi \right\Vert . \end{aligned}$$

Note that both $A(D_{\Gamma } - \lambda ) - I$ and $[\eta ,D_{\Gamma }]$ are compact. We also observe that for any compact operator $K :L^2(\Gamma _x) \rightarrow L^2(\Gamma _x)$ we have $V_{\alpha }^nKV_{\alpha }^{-n} \rightarrow 0$ in the strong operator topology, since $V_{\alpha }^{-n} \rightarrow 0$ in the weak operator topology. Therefore, and using that $V_{\alpha }^n\eta '^{3} V_{\alpha }^{-n} \rightarrow I$, we obtain that the second term in (5.4) tends to zero as $n\rightarrow \infty$. We conclude that $\left\Vert \phi \right\Vert \le \left\Vert A\right\Vert \left\Vert (D_{\Gamma _x} - \lambda )\phi \right\Vert$ for every $\phi \in L^2(\Gamma _x)$. The same argument also shows that $\left\Vert \phi \right\Vert \le \left\Vert A\right\Vert \left\Vert (D_{\Gamma _x} - \lambda )^\prime \phi \right\Vert$ for every $\phi \in L^2(\Gamma _x)$. It follows that $D_{\Gamma _x} - \lambda$ is invertible and thus Fredholm.

Conversely, choose $x_1,\ldots ,x_N \in \Gamma$ such that $\Gamma \subset \cup _{i=1}^N B(x_i, \delta (x_i))$, and suppose that $D_{\Gamma _{x_j}} - \lambda$ is Fredholm for $j=1,\ldots ,N$, and let $A_{j}$ be a Fredholm regularizer of $D_{\Gamma _{x_j}} - \lambda$. For every j let $\psi _{x_j} :\mathbb {R}^d \rightarrow \mathbb {R}^d$ be an isometry with $\psi _{x_j}(x_j) = 0$ and $\psi _{x_j}(B(x_j,\delta (x_j)) \cap \Gamma ) = B(0,\delta (x_j)) \cap \Gamma _{x_j}$. Choose Lipschitz continuous functions $\eta _1,\ldots ,\eta _N$ such that $\left\{ \eta _j^2: j = 1,\ldots ,N\right\}$ is a partition of unity of $\Gamma$ subordinate to the sets $B(x_j,\delta (x_j))$. Now note that, since $D_{\Gamma _{x_j}}$ essentially commutes with $\eta _j'$, $A_j$ essentially commutes with $\eta _j'$ as well. This implies that

$$\begin{aligned}&\sum \limits _{j = 1}^N \Psi _{x_j}\eta _j'A_{j}\Psi _{x_j}^\prime \eta _j(D_{\Gamma } - \lambda )\\&\quad \sim \sum \limits _{j = 1}^N \Psi _{x_j} A_{j}\eta _j'\Psi _{x_j}^\prime (D_{\Gamma } - \lambda )\eta _j = \sum \limits _{j = 1}^N \Psi _{x_j} A_{j}\Psi _{x_j}^\prime \eta _j(D_{\Gamma } - \lambda )\eta _j\\&\quad = \sum \limits _{j = 1}^N \Psi _{x_j} A_{j}\eta _j'(D_{\Gamma _{x_j}} - \lambda )\eta _j'\Psi _{x_j}^\prime \sim \sum \limits _{j = 1}^N \Psi _{x_j}\eta _j'A_{j}(D_{\Gamma _{x_j}} - \lambda )\eta _j'\Psi _{x_j}^\prime \\&\quad \sim \sum \limits _{j = 1}^N \Psi _{x_j}\eta _j'^2\Psi _{x_j}^\prime = \sum \limits _{j = 1}^N \eta _j^2 = I. \end{aligned}$$

Similarly, $(D_{\Gamma } - \lambda )\sum \limits _{j = 1}^N \eta _j\Psi _{x_j}A_{j}\eta _j'\Psi _{x_j}^\prime \sim I$. Thus $D_{\Gamma } - \lambda$ is Fredholm. $\square$

5.2 Deformations

Let $\beta \in (0,1)$. We will now consider the situation where a Lipschitz domain $\Omega _-$ with boundary $\Gamma$ is deformed by a $C^{1,\beta }$-diffeomorphism $\psi :\mathbb {R}^d \rightarrow \mathbb {R}^d$ that is conformal or anti-conformal at a point $x \in \Gamma$, meaning that the Jacobian $J\psi (x)$ of $\psi$ at x lies in ${\mathbb {R}} O(d)$, where ${\mathbb {R}} O(d):= \left\{ \lambda A \in \mathbb {R}^{d \times d}: \lambda \in \mathbb {R}, A \text { orthogonal}\right\}$. Due to the invariance of the double-layer potential under transformations from ${\mathbb {R}}O(d)$, we will for our purposes be able to assume that $J\psi (x) = I$. Let $D_\Gamma (\cdot ,\cdot )$ denote the kernel of $D_{\Gamma }$. Likewise, $V_j$ will be the integral operator with kernel $V_j(\cdot ,\cdot )$, $|D_{\Gamma }|$ will be the integral operator with kernel $|D_{\Gamma }(\cdot ,\cdot )|$, and so on. We write

$$\begin{aligned} D_{\psi (\Gamma )}(\psi (z), \psi (y))&= D_\Gamma (z,y) \nonumber \\&\quad + \underbrace{D_{\psi (\Gamma )}(\psi (z), \psi (y)) - \frac{(z-y) \cdot n^\Gamma (y)}{c_d|\psi (z) - \psi (y)|^d} n^{\psi (\Gamma )}(\psi (y)) \cdot J\psi (y) n^\Gamma (y)}_{V_1(z,y)}\nonumber \\&\quad +\, \underbrace{D_\Gamma (z,y) (n^{\psi (\Gamma )}(\psi (y)) \cdot J\psi (y) n^\Gamma (y) - 1)}_{V_2(z,y)} \nonumber \\&\quad + \underbrace{D_\Gamma (z,y) n^{\psi (\Gamma )}(\psi (y)) \cdot J\psi (y) n^\Gamma (y) \left[ \frac{|z-y|^d}{|\psi (z) - \psi (y)|^d} - 1 \right] }_{V_3(z,y)}. \end{aligned}$$

(5.5)

The term $V_1$ is weakly singular, satisfying an estimate of the form $|V_1(z,y)| \le C_{\beta } |z-y|^{1+\beta -d}$ for $\beta \in (0,1)$ and almost every $z,y \in \Gamma$, see [44, Lemma 6]. (In order to apply Medková’s results from Medková [44], note that the reduced boundary $\partial _*\Omega _- \subset \Gamma$ has full surface measure for a Lipschitz domain $\Omega _-$, cf. [17, p. 209].) Therefore the operator $V_1 :L^2(\Gamma ) \rightarrow L^2(\Gamma )$ is compact.

We view $V_2$ as the composition of $D_\Gamma$ with a multiplication operator, noting that by Medková [44, Lemma 3] we have

$$\begin{aligned} \lim _{\rho \rightarrow 0^+} \mathop {\mathrm {ess \, sup}}\limits \limits _{y \in \Gamma \cap B(x, \rho )} |n^{\psi (\Gamma )}(\psi (y)) \cdot J\psi (y) n^\Gamma (y) - 1| = 0. \end{aligned}$$

In particular, for every $\varepsilon > 0$, there is $\rho > 0$ such that the operator norm of

$$\begin{aligned} V_2 :L^2(\Gamma \cap B(x,\rho )) \rightarrow L^2(\Gamma ) \end{aligned}$$

is smaller than $\varepsilon$.

For a bounded Lipschitz domain with boundary $\Gamma$, let $\mathscr {A}_2 = \mathscr {A}_2(\Gamma )$ denote the algebra of integral operators K such that $|K| :L^2(\Gamma ) \rightarrow L^2(\Gamma )$ is bounded. We will work within the class of domains such that $D_\Gamma \in \mathscr {A}_2$ in this subsection. The preceding considerations show that $D_{\psi (\Gamma )} \in \mathscr {A}_2(\psi (\Gamma ))$ if $D_\Gamma \in \mathscr {A}_2(\Gamma )$. Since the relationship between $\Gamma$ and $\psi (\Gamma )$ is symmetric, we see that

$$\begin{aligned} D_{\psi (\Gamma )} \in \mathscr {A}_2(\psi (\Gamma )) \text { if and only if } D_\Gamma \in \mathscr {A}_2(\Gamma ). \end{aligned}$$

(5.6)

Note that $D_\Gamma \in \mathscr {A}_2$ for any $C^{1,\beta }$-boundary $\Gamma$, since $D_\Gamma$ has a weakly singular kernel in this case. It is also known that $|D_{\widetilde{\Gamma }}| :L^2(\widetilde{\Gamma }) \rightarrow L^2(\widetilde{\Gamma })$ is bounded for any of the following graphs $\widetilde{\Gamma }$:

• (i) a 2D wedge (modelling a polygonal corner) [49];

• (ii) a 3D wedge (modelling a polyhedral edge) [52];

• (iii) a 3D polyhedral cone [16];

• (iv) a 3D smooth cone (such as a circular cone) [53].

Furthermore, this statement may be extended to dilation-invariant domains which are built from such graphs.

Lemma 5.3

Suppose that $\widetilde{\Gamma }$ is a dilation-invariant graph, i.e., $\widetilde{\Gamma } = \alpha \widetilde{\Gamma }$, for some $\alpha \in (0,1)$, and suppose that, at every $x \in \widetilde{\Gamma } {\setminus } \left\{ 0\right\}$, $\widetilde{\Gamma }$ is either locally $C^{1,\beta }$ for some $\beta \in (0,1)$, or coincides locally with a graph of type (i)–(iii), or (iv). Then $|D_{\widetilde{\Gamma }}|$ is bounded on $L^2(\widetilde{\Gamma })$.

Proof

By hypothesis, we know that $\left|D_{j}\right| :L^2(\widetilde{\Gamma }_0) \rightarrow L^2(\widetilde{\Gamma }_0)$ is bounded for $j=-1,0,1$, where $D_j$ and $\widetilde{\Gamma }_0$ are as in Sect. 4.1. The lemma therefore follows from Proposition 4.3 and (4.4). $\square$

A simple localization argument now demonstrates that the condition $D_\Gamma \in \mathscr {A}_2$ is quite general.

Theorem 5.4

Suppose that at every $x \in \Gamma$, $\Gamma$ is locally $C^{1,\beta }$, for some $\beta \in (0,1)$, or coincides locally with a graph satisfying the hypothesis of Lemma 5.3. Then $D_\Gamma \in \mathscr {A}_2(\Gamma )$.

Returning to the decomposition of $D_{\psi (\Gamma )}(\psi (z), \psi (y))$, we note, for the third term $V_3$, the following consequence of the fact that $J\psi (x) = I$.

Lemma 5.5

Suppose that $D_\Gamma \in \mathscr {A}_2$. Then $|V_3|(z,y) = |D_\Gamma |(z,y) Q(z,y)$, where $Q \in L^\infty (\Gamma \times \Gamma )$ is non-negative and $\lim _{\rho \rightarrow 0^+} \mathop {\mathrm {ess \, sup}}\limits _{y,z \in \Gamma \cap B(x, \rho ) } Q(z,y) = 0$, where the essential supremum is taken with respect to the product measure on $\Gamma \times \Gamma$.

Let $\eta _\rho :\Gamma \rightarrow [0,1]$ be a Lipschitz continuous function with ${{\,\textrm{supp}\,}}\eta _{\rho } \subset B(x,\rho )$ such that $\eta _\rho$ is equal to 1 in a neighbourhood of x. Furthermore, denote the operator of composition with $\psi$ by

$$\begin{aligned} C_{\psi } :L^2(\psi (\Gamma )) \rightarrow L^2(\Gamma ), \quad C_\psi \phi (z) = \phi (\psi (z)), \end{aligned}$$

and let $\eta _{\rho }' = \eta _{\rho } \circ \psi ^{-1}$. Note also that we have the change of variables formula

$$\begin{aligned} \int _{\psi (\Gamma )} \phi (y) \, \textrm{d}s_{\psi (\Gamma )}(y) = \int _\Gamma (1+T(y))\phi (\psi (y)) \, \textrm{d}s_{\Gamma }(y), \end{aligned}$$

(5.7)

where $T \in L^\infty (\Gamma )$ and $\lim \nolimits _{y \rightarrow x} T(y) = 0$. The operator

$$\begin{aligned} \eta _{\rho }' D_{\psi (\Gamma )} \eta _{\rho }' :L^2(\psi (\Gamma )) \rightarrow L^2(\psi (\Gamma )) \end{aligned}$$

is then similar to

$$\begin{aligned} C_{\psi } \eta _{\rho }' D_{\psi (\Gamma )} \eta _{\rho }' C_{\psi ^{-1}} :L^2(\Gamma ) \rightarrow L^2(\Gamma ), \end{aligned}$$

which, by (5.7), is an integral operator with kernel $(1+T(y)) \eta _{\rho }(z) D_{\psi (\Gamma )}(\psi (z), \psi (y)) \eta _{\rho }(y)$. By the previous calculations, if $D_\Gamma \in \mathscr {A}_2$, we therefore find that

$$\begin{aligned} C_{\psi } \eta _{\rho }' D_{\psi (\Gamma )} \eta _{\rho }' C_{\psi ^{-1}} \sim \eta _{\rho } D_\Gamma \eta _{\rho } + \eta _{\rho }\tilde{V}\eta _{\rho }, \end{aligned}$$

(5.8)

where $\tilde{V}$ is an integral operator induced by a kernel of the form $\tilde{V}(z,y) = D_{\Gamma }(z,y)\tilde{Q}(z,y)$ with $\tilde{Q} \in L^\infty (\Gamma \times \Gamma )$ such that

$$\begin{aligned} \lim _{\rho \rightarrow 0^+} \mathop {\mathrm {ess \, sup}}\limits \limits _{y,z \in \Gamma \cap B(x, \rho )} |\tilde{Q}(z,y)| = 0. \end{aligned}$$

(5.9)

For $\phi _1,\phi _2 \in L^2(\Gamma )$ with support in $B(x,\rho )$ we have

$$\begin{aligned} \left|\left\langle \tilde{V}\phi _1, \phi _2 \right\rangle \right|\le & {} \int _{\Gamma } \int _{\Gamma } \left|\phi _2(z)\tilde{V}(z,y)\phi _1(y)\right| \, \textrm{d}s(y) \, \textrm{d}s(z) \\\le & {} \mathop {\mathrm {ess \, sup}}\limits \limits _{y,z \in \Gamma \cap B(x, \rho )} |\tilde{Q}(z,y)|\left\Vert \left|D_{\Gamma }\right|\right\Vert \left\Vert \phi _1\right\Vert \left\Vert \phi _2\right\Vert . \end{aligned}$$

Since $D_\Gamma \in \mathscr {A}_2$, (5.9) implies that

$$\begin{aligned} \lim _{\rho \rightarrow 0^+} \Vert \eta _{\rho } \tilde{V}\eta _{\rho }\Vert = 0. \end{aligned}$$

(5.10)

Taking powers, we conclude that

$$\begin{aligned} \lim _{\rho \rightarrow 0^+} \Vert (\eta _{\rho } D_\Gamma \eta _{\rho })^n - C_{\psi } (\eta _{\rho }' D_{\psi (\Gamma )} \eta _{\rho }')^nC_{\psi ^{-1}}\Vert _{{{\,\textrm{ess}\,}}} = 0 \end{aligned}$$

(5.11)

for every $n \ge 1$.

We now prove the main theorem of this section. When $R = 1/2$, it says the following: if, for every point x, $\Gamma$ is locally obtained by deforming a domain that satisfies the spectral radius conjecture, then $\Gamma$ satisfies the spectral radius conjecture as well.

Theorem 5.6

Let $R > 0$ and $\beta > 0$. Suppose that for every $x \in \Gamma$ there exists a $C^{1,\beta }$-diffeomorphism $\psi _x :{\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$, conformal or anti-conformal at x, and a Lipschitz cut-off function $\eta _x :\mathbb {R}^d \rightarrow [0,1]$ such that $\eta _x$ is equal to 1 in a neighbourhood of x, $D_{\psi _x(\Gamma )} \in \mathscr {A}_2$, and

$$\begin{aligned} \sigma _{{{\,\textrm{ess}\,}}}(\eta _x' D_{\psi _x(\Gamma )}\eta _x') \subset B(0,R), \end{aligned}$$

where $\eta _x' = \eta _x \circ \psi _x^{-1}$. Then $D_\Gamma \in \mathscr {A}_2$ and $\sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma }) \subset B(0,R)$.

Proof

As noted previously, the condition on $\Gamma$, even for just one x, implies that $D_\Gamma \in \mathscr {A}_2$.

Fix $x \in \Gamma$. Since the commutator $[D_{\psi _x(\Gamma )}, \eta _x']$ is compact, we see that $(\eta _x' D_{\psi _x(\Gamma )}\eta _x')^n - \eta _x'^n D_{\psi _x(\Gamma )}^n \eta _x'^n$ is compact for every $n \ge 1$. By the hypothesis, choose $n = n(x)$ sufficiently large so that

$$\begin{aligned} \Vert (\eta _x' D_{\psi _x(\Gamma )}\eta _x')^n\Vert _{{{\,\textrm{ess}\,}}}^{\frac{1}{n}} = \Vert \eta _x'^n D_{\psi _x(\Gamma )}^n\eta _x'^n\Vert _{{{\,\textrm{ess}\,}}}^{\frac{1}{n}} < R\Vert C_{\psi _x}\Vert ^{-\frac{1}{n}}\Vert C_{\psi _x^{-1}}\Vert ^{-\frac{1}{n}}. \end{aligned}$$

If we choose a new cut-off function $\tilde{\eta }_x :\mathbb {R}^d \rightarrow [0,1]$ such that ${{\,\textrm{supp}\,}}\tilde{\eta }_x \subset \left\{ y \in \mathbb {R}^d: \eta _x(y) = 1\right\}$, note that, since $(\eta _x' D_{\psi _x(\Gamma )}\eta _x')^n - \eta _x'^n D_{\psi _x(\Gamma )}^n \eta _x'^n$ is compact, we have

$$\begin{aligned}{} & {} \Vert (\tilde{\eta }_x' D_{\psi _x(\Gamma )}\tilde{\eta }_x')^n\Vert _{{{\,\textrm{ess}\,}}}^{\frac{1}{n}} = \Vert \tilde{\eta }_x'^n D_{\psi _x(\Gamma )}^n\tilde{\eta }_x'^n\Vert _{{{\,\textrm{ess}\,}}}^{\frac{1}{n}}\\ {}{} & {} \le \Vert \eta _x'^n D_{\psi _x(\Gamma )}^n\eta _x'^n\Vert _{{{\,\textrm{ess}\,}}}^{\frac{1}{n}} < R\Vert C_{\psi _x}\Vert ^{-\frac{1}{n}}\Vert C_{\psi _x^{-1}}\Vert ^{-\frac{1}{n}}, \end{aligned}$$

so that

$$\begin{aligned} \Vert C_{\psi _x} (\tilde{\eta }_x' D_{\psi _x(\Gamma )} \tilde{\eta }_x' )^nC_{\psi _x^{-1}} \Vert _{{{\,\textrm{ess}\,}}}^{\frac{1}{n}} \le \Vert \eta _x'^n D_{\psi _x(\Gamma )}^n\eta _x'^n \Vert _{{{\,\textrm{ess}\,}}}^{\frac{1}{n}}\Vert C_{\psi _x}\Vert ^{\frac{1}{n}}\Vert C_{\psi _x^{-1}}\Vert ^{\frac{1}{n}} < R. \end{aligned}$$

Hence, by making the support of $\eta _x$ sufficiently small and applying (5.11), we obtain that

$$\begin{aligned} \Vert (\eta _x D_\Gamma \eta _x)^n\Vert _{{{\,\textrm{ess}\,}}}^{\frac{1}{n}} = \Vert \eta _x^n D_\Gamma ^n \eta _x^n\Vert _{{{\,\textrm{ess}\,}}}^{\frac{1}{n}} < R. \end{aligned}$$

Now choose $\delta (x)$ sufficiently small so that $\eta _x \equiv 1$ in a neighbourhood of $B(x, \delta (x)) \cap \Gamma$. By compactness, we may pick points $x_1, \ldots , x_N$ so that $\Gamma \subset \cup _{j=1}^N B(x_j, \delta (x_j))$. Let

$$\begin{aligned} r:= \max _{j = 1,\ldots , N} \Vert (\eta _{x_j} D_\Gamma \eta _{x_j})^{n(x_j)}\Vert _{{{\,\textrm{ess}\,}}}^{\frac{1}{n(x_j)}} < R \end{aligned}$$

and let $\left\{ \tau _j: j = 1,\ldots , N\right\}$ be a Lipschitz partition of unity of $\Gamma$ subordinate to the sets $B(x_j, \delta (x_j))$. Choose m to be a large integer such that $n(x_j)$ divides m for all j. Then

$$\begin{aligned} D^m_\Gamma = \sum \limits _{j = 1}^N \tau _j D_\Gamma ^m = \sum _{j=1}^{N} \tau _j \eta _{x_j}^{2m} D_\Gamma ^m \end{aligned}$$

and therefore, recalling that $[D_\Gamma , \eta _{x_j}]$ is compact,

$$\begin{aligned} \Vert D_\Gamma ^m\Vert _{{{\,\textrm{ess}\,}}}&= \Bigg \Vert \sum \limits _{j = 1}^N \tau _j (\eta _{x_j} D_\Gamma \eta _{x_j})^m\Bigg \Vert _{{{\,\textrm{ess}\,}}} \le \sum \limits _{j = 1}^N \left\Vert (\eta _{x_j} D_\Gamma \eta _{x_j})^m\right\Vert _{{{\,\textrm{ess}\,}}}\\ {}&\le \sum \limits _{j = 1}^N \left\Vert (\eta _{x_j} D_\Gamma \eta _{x_j})^{n(x_j)}\right\Vert ^{\frac{m}{n(x_j)}}_{{{\,\textrm{ess}\,}}}\\&\le Nr^m < N R^m. \end{aligned}$$

Therefore $\sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma }) \subset B(0,R)$. $\square$

Referring back to Lemma 5.3, all graphs $\widetilde{\Gamma }$ of type (i)–(iii) satisfy (the graph version of) the spectral radius conjecture, results which can also be found in the corresponding references. Graphs of type (iv) that are convex also satisfy the spectral radius conjecture, by Fabes et al. [18]. If $\eta :{\mathbb {R}}^3 \rightarrow [0,1]$ is a compactly supported Lipschitz function, we thus have that

$$\begin{aligned} \Vert (\eta D_{\widetilde{\Gamma }} \eta )^n \Vert _{{{\,\textrm{ess}\,}}} = \Vert \eta ^n D_{\widetilde{\Gamma }}^n \eta ^n \Vert _{{{\,\textrm{ess}\,}}} \le \Vert D_{\widetilde{\Gamma }}^n\Vert _{{{\,\textrm{ess}\,}}} < 2^{-n} \end{aligned}$$

for n sufficiently large. Applying Theorem 5.6 we obtain the following.

Corollary 5.7

Let $d = 3$ and let $\Gamma$ be the boundary of a Lipschitz domain. Assume that for every $x \in \Gamma$ there exists a $C^{1,\beta }$-diffeomorphism $\psi _x :{\mathbb {R}}^3 \rightarrow {\mathbb {R}}^3$, conformal or anti-conformal at x, such that $\psi _x(\Gamma )$ is locally a subset of the boundary of a polyhedral cone or a convex smooth cone. Then $D_\Gamma$ satisfies the spectral radius conjecture.

The class of domains considered in Corollary 5.7 encompasses all domains in 3D that may reasonably be referred to as a Lipschitz curvilinear polyhedra. From Theorem 5.6 we of course also obtain the analogous corollary for 2D domains, a well known result; the corresponding class of curves precisely describes the $C^{1,\beta }$-curvilinear polygons in 2D.

5.3 Possible extensions and further questions

Theorem 5.6 in particular shows that the spectral radius conjecture only depends on the local behaviour of $\Gamma$, under the extraneous assumption that $D_\Gamma \in \mathscr {A}_2$. To achieve the same result for a general Lipschitz domain, we need a different tool to treat the term $V_3$ which does not rely on the introduction of absolute values. Since $V_3(z,y) = \widetilde{Q}(z,y) D_\Gamma (z,y)$, for a specific kernel $\widetilde{Q}(z,y)$, we would like an answer to the following question.

Question 5.8

Describe suitable classes of kernels B such that the kernel $B(z,\!\!y) D_\Gamma (z,\!\!y)$ defines a bounded operator on $L^2(\Gamma )$, and estimate the norm of the corresponding operator.

When B is of the form $B(z,\!y) = a(z) - a(y)$, for some function a on $\Gamma$, the resulting operator is the commutator $[a, D_\Gamma ]$. Commutators of singular integral operators are very well studied, but we have been unable to identify or correctly apply existing results to the kernel multiplier of interest to us, namely,

$$\begin{aligned} B(z,y) = \eta _\rho (z) \left( \frac{|z-y|^d}{|\psi (z) - \psi (y)|^d} - 1\right) \eta _\rho (y). \end{aligned}$$

Other versions of Question 5.8 also seem interesting. For example, one could ask for the stronger property that B(z, y) be a kernel multiplier of each of the Riesz transforms of $\Gamma$. We also note the similarity between the term $V_3$ and the kernel of the Clifford–Cauchy integral operator, as presented in [4, Consequence 3.6].

Of course, one would like to know that not only the spectral radius conjecture is local, but that the entire essential spectrum is as well.

Question 5.9

For a domain $\Omega _-$ with boundary $\Gamma$, is the essential spectrum local and invariant under locally conformal deformations?

We can give a positive answer to this question if we, in addition to the hypotheses of Theorem 5.6, assume that we do not have too many singular points, by which we mean points where the boundary is not $C^1$. In the statement, we let $\chi _{x,\rho }:= \chi _{B(x,\rho )}$ and $\chi _{x,\rho }':= \chi _{x,\rho } \circ \psi _x^{-1}$ for $x \in \Gamma$ and $\rho > 0$.

Theorem 5.10

Let $\beta > 0$ and assume that $\Gamma$ has at most countably many singular points. Further suppose that for every $x \in \Gamma$ there exists a $C^{1,\beta }$-diffeomorphism $\psi _x :{\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$, conformal or anti-conformal at x, such that $D_{\psi _x(\Gamma )} \in \mathscr {A}_2$. Then $D_\Gamma \in \mathscr {A}_2$ and

$$\begin{aligned} \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma }) \subset \bigcup \limits _{x \in \Gamma } \sigma _{{{\,\textrm{ess}\,}}}(\chi _{x,\rho _x}'D_{\psi _x(\Gamma )}\chi _{x,\rho _x}') \end{aligned}$$

for arbitrary $\rho _x > 0$. Moreover, there exist $\rho _x > 0$ such that

$$\begin{aligned} \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma }) = \bigcup \limits _{x \in \Gamma } \sigma _{{{\,\textrm{ess}\,}}}(\chi _{x,\rho _x}'D_{\psi _x(\Gamma )}\chi _{x,\rho _x}'). \end{aligned}$$

Proof

By (5.6), $D_{\psi _x(\Gamma )} \in \mathscr {A}_2$ for just one $x \in \Gamma$ already implies $D_\Gamma \in \mathscr {A}_2$.

Let $x \in \Gamma$, $\rho _x > 0$ and $\lambda \in \mathbb {C}$. If $\lambda = 0$, then $0 \in \sigma _{{{\,\textrm{ess}\,}}}(\chi _{x,\rho _x}'D_{\psi _x(\Gamma )}\chi _{x,\rho _x}')$ either by (5.1) (if $\chi _{x,\rho _x}' \equiv 1$) or by construction (otherwise). So assume that $\lambda \ne 0$ and suppose that $\chi _{x,\rho _x}'D_{\psi _x(\Gamma )}\chi _{x,\rho _x}' - \lambda$ is Fredholm for every $x \in \Gamma$.

Fix x for a moment. As $\Gamma$ has at most countably many singular points and $\psi _x$ is a diffeomorphism, we know that $\chi _{x,\tilde{\rho }_x}'D_{\psi _x(\Gamma )}\chi _{x,\tilde{\rho }_x}' - \lambda$ is Fredholm for all but at most countably many $\tilde{\rho }_x \in (0,\rho _x]$. In this case the essential norm of the Fredholm regularizer $A_{x,\tilde{\rho }_x}$ of $\chi _{x,\tilde{\rho }_x}'D_{\psi _x(\Gamma )}\chi _{x,\tilde{\rho }_x}' - \lambda$ is bounded by

$$\begin{aligned} \left\Vert A_{x,\tilde{\rho _x}}\right\Vert _{{{\,\textrm{ess}\,}}} \le \max \left\{ \left\Vert A_{x,\rho _x}\right\Vert _{{{\,\textrm{ess}\,}}},\left|\lambda \right|^{-1}\right\} . \end{aligned}$$

Therefore, by (5.8) and (5.10), for every $x \in \Gamma$ we can choose $\tilde{\rho }_x$ sufficiently small such that $\chi _{x,\tilde{\rho }_x}D_{\Gamma }\chi _{x,\tilde{\rho }_x} - \lambda$ is also Fredholm, and such that $\partial B(x, \tilde{\rho }_x)$ contains none of the singularities of $\Gamma$.

By compactness, we may choose finitely many points $x_j$ and corresponding radii $\tilde{\rho }_{x_j}$ such that the balls $B(x_j,\tilde{\rho }_{x_j})$ cover $\Gamma$. Define recursively

$$\begin{aligned} E_1:= B(x_1,\tilde{\rho }_{x_1}) \cap \Gamma , \quad E_{j+1} = \big (B(x_{j+1},\tilde{\rho }_{x_{j+1}}) \cap \Gamma \big ) \setminus \bigcup \limits _{k = 1}^j E_k. \end{aligned}$$

Now we can apply (5.2) to obtain that $D_{\Gamma } - \lambda$ is Fredholm.

Conversely, assume that $D_{\Gamma } - \lambda$ is Fredholm and fix $x \in \Gamma$. By (5.1), we must have $\lambda \ne 0$. As we only have countably many singularities, $\chi _{x,\rho }D_{\Gamma }\chi _{x,\rho } - \lambda$ is also Fredholm for all but at most countably many $\rho > 0$ by (5.2). Now note that the situation between $D_{\Gamma }$ and $D_{\psi _x(\Gamma )}$ is symmetric. Hence, the same argument as above shows that $\chi _{x,\rho _x}'D_{\psi _x(\Gamma )}\chi _{x,\rho _x}' - \lambda$ is also Fredholm for a sufficiently small $\rho _x > 0$. $\square$

In particular, we may extend Theorem 5.2 to domains that are only approximately locally dilation invariant in the following sense.

Corollary 5.11

Let $\beta > 0$ and assume that $\Gamma$ has at most countably many singular points. Denote the set of singular points by $\mathcal {S}$. Suppose that for every $x \in \mathcal {S}$ there exists a $C^{1,\beta }$-diffeomorphism $\psi _x :{\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$ such that

• (i) $\psi _x$ is conformal or anti-conformal at x,

• (ii) $\psi _x(\Gamma )$ is locally $C^{1,\beta }$ or locally dilation invariant at $\psi _x(x)$,

• (iii) $D_{\psi _x(\Gamma )} \in \mathscr {A}_2$ (e.g., a domain with local behaviour as in Theorem 5.4).

Then there exist $\rho _x > 0$ such that $\sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma }) = \left\{ 0\right\} \cup \bigcup \nolimits _{x \in \mathcal {S}} \sigma _{{{\,\textrm{ess}\,}}}(\chi _{x,\rho _x}'D_{\psi _x(\Gamma )}\chi _{x,\rho _x}')$.

6 Synthesis and numerical examples

In this final section we bring earlier results together to study the case where $\Gamma \in \mathscr {D}_\mathcal {A}$ (recall Definition 1.4), meaning that $\Gamma$ is the boundary of a bounded Lipschitz domain $\Omega _-\subset \mathbb {R}^2$ that is locally dilation invariant and is piecewise analytic. We then illustrate the various results of the paper by several examples. Regarding our first aim, the following result is immediate from Theorems 5.2, 4.19, and 4.21.

Theorem 6.1

Suppose that $\Gamma \in \mathscr {D}_\mathcal {A}$ and let $F\subset \Gamma$ be the finite set of points at which $\Gamma$ is not locally analytic but is locally dilation invariant, so that $\Gamma$ coincides locally near $x\in F$ with $\Gamma _x$, a dilation invariant graph (in some rotated coordinate system centred at x). Then, where $\sigma ^N$ is as defined in Theorem 4.19, as $N\rightarrow \infty$. Further, suppose that $c>0$ is small enough such that the conditions of the first sentence of Theorem 4.21 are satisfied for $\Gamma = \Gamma _x$, $x\in F$. Then, for every $\rho _0>0$, where $R_N(D_\Gamma )$ is defined by (1.13), $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma ))< \rho _0$ if: i) $R_N(D_\Gamma )< \rho _0$; and ii) for some $m,M,N\in \mathbb {N}$, with $M\ge 2$,

$$\begin{aligned} \mathscr {S}_c(\Gamma ,m,M,N) := \max _{x\in F} S_c(\Gamma _x,m,M,N) < 0, \end{aligned}$$

(6.1)

where $S_c(\cdot ,\cdot ,\cdot ,\cdot )$ is as defined in Remark 4.22. Conversely, if $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma ))< \rho _0$, then, for all sufficiently large N, $R_N(D_\Gamma )<\rho _0$, and, if $m, M\in \mathbb {N}$ are also sufficiently large, then $\mathscr {S}_c(\Gamma ,m,M,N)<0$.

Proof

Since $\Gamma \in \mathscr {D}_\mathcal {A}\subset \mathscr {D}$, $\Gamma$ is locally $C^1$ at x if $x\in \Gamma \setminus F$, while, if $x\in F$, $\Gamma _x$ satisfies (4.44) (in some local coordinate system centred at x). Thus, and by Theorem 5.2, $\sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma )= \cup _{x\in \Gamma } \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _x}) = \cup _{x\in F}\sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _x})$, since $\sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _x})=\{0\}$ if $x\in \Gamma {\setminus } F$, and $0\in \sigma _{{{\,\textrm{ess}\,}}}(D_{\Gamma _x})$ if $x\in F$, by Theorem 4.4 and Corollary 4.6. The result thus follows from Theorems 4.19 and 4.21. $\square$

The following examples illustrate the above result and the results of Sects. 4 and 5. In each example, whether $\Gamma$ is a dilation invariant graph or $\Gamma \in \mathscr {D}_\mathcal {A}$, we demonstrate that $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma ;L^2(\Gamma ))<\frac{1}{2}$, providing new evidence in support of the spectral radius conjecture.

Example 1

We first consider an example where $\Gamma$ satisfies (4.8), with $f:\mathbb {R}_+\rightarrow \mathbb {R}$ given by

$$\begin{aligned} f(x) := x\sin ^2(\pi \log _{\alpha }(x)), \quad x>0, \end{aligned}$$

(6.2)

for some $\alpha \in (0,1)$, so that [recall (4.10)] $g(x) = \sin ^2(\pi x) = (1-\cos (2\pi x))/2$, for $x\in \mathbb {R}$. As $\alpha$ increases in (0, 1) the graph of f (see Fig. 2) becomes increasingly oscillatory and its Lipschitz character increases; elementary calculations give that the maximum and minimum of $f^\prime$ are $f^\prime _{\max }=\cos ^2(\theta _\alpha /2) +\pi \sin (\theta _\alpha )/|\log \alpha |>1$ and $f^\prime _{\min }=1-f^\prime _{\max }<0$, where $\theta _\alpha := \arctan (2\pi /|\log \alpha |)$; note that $f^\prime _{\max } = \pi /|\log \alpha | + 1/2 + O(1-\alpha )$ as $\alpha \rightarrow 1^-$. To apply Theorem 4.15 we need to compute $C_1(M)$, $C_3$, and $C_4$; see (4.34), (4.39), and the definition below (4.40). This requires computation of quantities that are defined in Proposition 4.8 in terms of²⁰

$$\begin{aligned} \Vert {{\,\textrm{Im}\,}}g^{(j)}\Vert _c = \frac{(2\pi )^j}{2} \sinh (2\pi c), \quad j = 0,1, \quad c\ge 0,\\ \Vert g\Vert _c = \cosh ^2(\pi c), \quad \Vert g^{(j)}\Vert _c = \frac{(2\pi )^j}{2} \cosh (2\pi c), \quad j = 1,2,\quad c\ge 0. \end{aligned}$$

Theorem 4.15 applies for all $c>0$ such that (4.21) applies, i.e. provided

$$\begin{aligned} c \le \frac{\arccos (\alpha )}{\left|\log \alpha \right|} \quad \text {and} \quad c < \frac{1}{2\pi }{{\,\textrm{arsinh}\,}}\left( \frac{2|\log \alpha |}{2\pi + |\log \alpha |}\right) . \end{aligned}$$

(6.3)

Fig. 4

Numerical results for Example 1; f given by (6.2), $\alpha =\frac{3}{4}$

In Fig. 4 we plot results for the case $\alpha =\frac{3}{4}$ (see Fig. 2), when $f^\prime _{\max } \approx 11.43$ and the above conditions reduce to $c \le \arccos (3/4)/\log (4/3) \approx 2.51$ and $c < {{\,\textrm{arsinh}\,}}\left( 2 \log (4/3)/(2\pi + \log (4/3))\right) /(2\pi )$ $\approx 0.0139$. Choosing $c=0.013$ we plot in Fig. 4a, for the case $\rho _0=\frac{1}{2}$, $\mathcal {L}_c(f,N)$ and $\mathcal {R}_c(f,m,M,N)$, given by (4.43), against N, for $N=2^j$, $j=3,4,\ldots ,9$, choosing $m=16,000$ and $M=100$. We see that with these choices $\mathcal {R}_c(f,m,M,N)$ is positive and bounded away from zero for sufficiently large N, while, as is clear from its definition, $\mathcal {L}_c(f,N)$ decreases exponentially with N; note that (4.43) is satisfied for $N=512$. To apply Theorem 4.15 to conclude that $\rho (D_\Gamma )=\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$ we also need to check that $\rho (A^M_{t,N})<\frac{1}{2}$, for $M=100$, N = 512, and $t = t_k=(k-1/2)/m$, $k=1,\ldots , m$, with $m=16,000$; equivalently, that $r^N_{\max }:= \max _{z\in \sigma ^N(D_\Gamma )}|z|<\frac{1}{2}$, where $\sigma ^N(D_\Gamma )$ is as defined by (4.41) with $N=512$, $m_N=16,000$, $M_N=100$. For these parameter values $\sigma ^N(D_\Gamma )$, an approximation to $\sigma (D_\Gamma )=\sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma )$ by Theorem 4.13, is plotted in Fig. 4b, and $r^{N}_{\max } \approx 0.4837 < \frac{1}{2}$. Our calculations are in standard double-precision floating-point arithmetic rather than exact arithmetic,²¹ but this is convincing evidence, by application of Theorem 4.15, that $\rho (D_\Gamma )<\frac{1}{2}$.

In Fig. 4b we also plot, for the parameter values $p=10$, $N=512$, $M=100$, $n=100$, and $t=\pi /18$, the bounded domain $W_n$, which²² is contained in and is an approximation to $W(T^{p,t,N,M})$ (these notations defined in Sect. 4.4). Where $R^*$ is as defined in Corollary 4.25, we also plot the circle $R^* \mathbb {T}$ of radius $R^*\approx 1.163$; by Corollary 4.25 it is guaranteed that $\overline{B_{R^*}}\subset W(D_\Gamma )=W_{{{\,\textrm{ess}\,}}}(D_\Gamma )$, so that $w(D_\Gamma )=w_{{{\,\textrm{ess}\,}}}(D_\Gamma )\ge R^*$ is significantly larger than $\frac{1}{2}$ for this example. We note that, for these parameter values, $C_7(p,N,M) \le 3.964 \times 10^{-5}$ (see Corollaries 4.24 and 4.25).

In Fig. 4b we additionally plot, to illustrate the application of Theorem 4.15 and the adaptive definition of the parameters defined by (4.42), the points $\mu _{k,\ell }$ at which the resolvent of $A_{t_k,N}^M$ is calculated when $\rho _0=\frac{1}{2}$, $N=512$, $M=100$, $m=16,000$, for the case $k=16,000$ (so that $t_k\approx \pi$) and $\ell =1,\ldots ,n_k=67$. This value of k is fairly typical; $n_k$ varies in the range [67, 110] as a function of k. It is clear that the adaptive algorithm of Theorem 4.15 (and see Corollary 3.5) is significantly more efficient than the uniform grid on $\rho _0 \mathbb {T}$ of Corollary 3.3 (recall the discussion above Lemma 3.4). For $k = 16,000$ we have $\min _\ell \nu _{k,\ell } \approx 0.0101$ and $\max _\ell \nu _{k,\ell } \approx 0.2216$ [see (4.42) for this notation]. The ratio of these maximum and minimum values, which is approximately the ratio of the maximum to the minimum spacing of the points $\mu _{k,\ell }$, is about 20.2.

Example 2

We now turn to examples where we can apply the theory of Sect. 4.3. First we consider the case of a cone, that is $\Gamma =\{(x,f(x)):x\in \mathbb {R}\}$ where

$$\begin{aligned} f(x) := \mu \left|x\right|, \quad x\in \mathbb {R}, \end{aligned}$$

(6.4)

for some $\mu \in \mathbb {R}$. This clearly satisfies (4.44), for any $\alpha \in (0,1)$, and, where $f_\pm$ and $g_\pm$ are defined by (4.45), $f_\pm (x) = \mu x$, $g_\pm (x)=\mu$, for $x>0$. For this example the spectrum and spectral radius are known, viz.

$$\begin{aligned}{} & {} \sigma _{{{\,\textrm{ess}\,}}}(\Gamma )=\sigma (D_\Gamma ) = \{0\}\cup \left\{ \pm \frac{\sin (\arctan (|\mu |)(1-iy))}{2\sin (\pi (1-iy)/2)}: y\in \mathbb {R}\right\} \nonumber \\ {}{} & {} \quad \text{ so } \text{ that } \;\; \rho (D_\Gamma ) = \frac{|\mu |}{2\sqrt{1+\mu ^2}} \end{aligned}$$

(6.5)

(see, e.g., [49]) and, since $D_\Gamma$ is diagonalised by the Mellin transform when $\Gamma$ is a 2D cone, $D_\Gamma$ is normal so that $W_{{{\,\textrm{ess}\,}}}(D_\Gamma )=\overline{W(D_\Gamma )}=\mathop {\textrm{conv}}\limits (\sigma (D_\Gamma ))$ and $\Vert D_\Gamma \Vert =w(D_\Gamma )=\rho (D_\Gamma )$.

To make comparison of these known results with the methods of Sect. 4.3 we can choose, in principle, any $\alpha \in (0,1)$, but the choice of $\alpha$ affects the choice of c via (4.54), and thus the rate of decrease with N of $\mathcal {L}_c(f,N)$, defined by (4.72). The operators $\tilde{L}^{\pm }_t$, given by (4.48), and thus the matrices $A_{t_k,N}^M$, given by (4.63), also depend on $\alpha$, while the operators $\tilde{K}^{\pm }_t$, given by (4.47), vanish in this case. It is worth noting that the blocks of $A_{t_k,N}^M$ corresponding to ${\tilde{L}}_t^\pm$ are Toeplitz matrices in this example. The conditions (4.54) reduce to $c\le \arccos (\alpha )/|\log \alpha |$ and $|\mu |c\le \alpha ^2/|\log \alpha |$. As the right hand sides of these inequalities are increasing on (0, 1), it is beneficial to choose $\alpha$ closer to 1 in order to be able to choose a larger c. We select $\alpha = \frac{7}{8}$, so that $\arccos (\alpha )/|\log \alpha | \approx 3.78$ and $\alpha ^2/|\log \alpha | \approx 5.73$, and then take $c=0.57$, so that (4.54) is satisfied for $|\mu |\le 10$. For $\mu =10$ we see from Fig. 5a that, with $M = 200$, $m = 2000$ and $N = 16$, the inequality (4.72) is satisfied and $\max _{k=1,\ldots ,m} \rho (A_{t_k,N}^M) \approx 0.4975 <\frac{1}{2}$, so that $\rho (D_\Gamma )<\frac{1}{2}$ by Theorem 4.21, in agreement with (6.5) which gives $\rho (D_\Gamma ) =5/\sqrt{101}\approx 0.4975$. We see in Fig. 5b that the approximations $\sigma ^N(D_\Gamma )$ to $\sigma (D_\Gamma )$, given by Theorem 4.19, agree closely, for the given parameter values, with the expected lemniscates given by (6.5) for different values of $\mu$.

Fig. 5

Numerical results for Example 3; f given by (6.4), $\alpha =\frac{7}{8}$

Example 3

In this example we take

$$\begin{aligned} f(x) := \left|x\right|\sin ^2(\pi \log _{\alpha }\left|x\right|), \quad x\in \mathbb {R}, \end{aligned}$$

(6.6)

and $\Gamma$ satisfies (4.44), but now just for one $\alpha \in (0,1)$. Note that $f_\pm$ and $g_\pm$, given by (4.45), are the same as f and g in Example 1, so that $g_\pm$ satisfy the same bounds as g in Example 1, and $\Gamma$ has Lipschitz character $f^\prime _{\max }$ as defined in that example. Thus the conditions (4.54) reduce in this example to (6.3) plus the condition that $\sinh (2\pi c)/2 + c|\log \alpha | \cosh ^2(\pi c) < \alpha ^2$. We choose $\alpha =\frac{2}{3}$ (see Fig. 3), for which $c=0.019$ satisfies this condition and (4.54), and $f_{\max }^\prime \approx 8.26$. We see from Fig. 6a that, with $M = 60$, $m = 10,000$ and $N = 256$, the inequality (4.72) is satisfied and $\max _{k=1,\ldots ,m} \rho (A_{t_k,N}^M)<\frac{1}{2}$, so that $\rho (D_\Gamma )<\frac{1}{2}$ by Theorem 4.21. Figure 6b plots an approximation $\sigma ^N(D_\Gamma )$ to $\sigma (D_\Gamma )$ given by Theorem 4.19, which is contained in the circle of radius $\frac{1}{2}$ (in red). By contrast, by (4.73), and arguing as in Example 1, the numerical range $W(D_\Gamma )$ contains at least the closed disc of radius $R^*\approx 0.8179$ (shown in blue in Fig. 6b), where $R^*$ is as given in Corollary 4.25.

Fig. 6

Numerical results for Example 3; f given by (6.6), $\alpha = \frac{2}{3}$

Figure 6 is also, by Theorem 6.1, relevant to the bounded Lipschitz domain $\Omega _-$ shown in Fig. 1b. This has boundary $\Gamma \in \mathscr {D}_\mathcal {A}$ which is $C^1$ except at the single point $x=0$ where it coincides locally with the graph of f given by (6.6). It follows by the above calculations and Theorem 6.1 that $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$ for $\Gamma =\partial \Omega _-$ and that Fig. 6b is also a plot of the approximation $\Sigma ^N(D_\Gamma )$, defined in Theorem 6.1, to the essential spectrum of $D_\Gamma$ for $\Gamma =\partial \Omega _-$. While $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$, $W_{{{\,\textrm{ess}\,}}}(D_\Gamma )$, for $\Gamma =\partial \Omega _-$, contains the closed disc of radius $R^*\approx 0.8179$ shown in Fig. 6b, by the above result for the graph $\Gamma$ given by (6.6), and a localisation result [11, Theorem 3.2] (and see [11, Theorem 3.17]) for the essential numerical range, analogous to Theorem 5.2. Thus also, by (1.8), $\Vert D_\Gamma \Vert _{{{\,\textrm{ess}\,}}}\ge w_{{{\,\textrm{ess}\,}}}(D_\Gamma )\ge R^*$.

Example 4

In this example we define f by (6.2) for $x>0$ and set $f(x):=0$ for $x\le 0$, so that $\Gamma$ and f satisfy (4.44). Recalling (4.45), we see that $f_+$ and $g_+$ are the same as f and g in Example 1, while $f_-=0$ and $g_-=0$. The graph $\Gamma$ has Lipschitz character $L= (f_{\max }^\prime -f_{\min }^\prime )/2 = f_{\max }^\prime - 1/2$, where $f_{\max }^\prime$ and $f_{\min }^\prime$ are as defined in Example 1. The conditions (4.54) reduce in this example to the same conditions as in Example 3, and again we choose $\alpha =\frac{2}{3}$ and $c=0.019$, so that $\Gamma$ has Lipschitz character $L=f_{\max }^\prime -1/2\approx 7.76$. Similarly to the previous example, we see from Fig. 7a that, with $M = 50$, $m = 5,000$, and $N = 256$, the inequality (4.72) is satisfied and $\max _{k=1,\ldots ,m} \rho (A_{t_k,N}^M)<\frac{1}{2}$, so that $\rho (D_\Gamma )<\frac{1}{2}$ by Theorem 4.21. The approximation $\sigma ^N(D_\Gamma )$ to $\sigma (D_\Gamma )$, given by Theorem 4.19 and plotted in Fig. 7b, is contained in the circle of radius $\frac{1}{2}$, while the numerical range $W(D_\Gamma )$ contains at least the closed disc of radius $R^*\approx 0.8179$ shown in Fig. 7b.

Figure 7 is also, by Theorem 6.1, relevant to the bounded Lipschitz domain $\Omega _-$ shown in Fig. 1a. This has boundary $\Gamma \in \mathscr {D}_\mathcal {A}$ which is $C^1$ except at the single point $x=0$ where it coincides locally with the graph of the function f described above. It follows by the above calculations and Theorem 6.1 that $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$ for $\Gamma =\partial \Omega _-$ and that Fig. 6b is also a plot of the approximation $\Sigma ^N(D_\Gamma )$, defined in Theorem 6.1, to the essential spectrum of $D_\Gamma$ for $\Gamma =\partial \Omega _-$. Arguing as in the previous example, while $\rho _{{{\,\textrm{ess}\,}}}(D_\Gamma )<\frac{1}{2}$, $W_{{{\,\textrm{ess}\,}}}(D_\Gamma )$, for $\Gamma =\partial \Omega _-$, contains the closed disc of radius $R^*\approx 0.8179$ plotted in Fig. 7b, and $\Vert D_\Gamma \Vert _{{{\,\textrm{ess}\,}}}\ge w_{{{\,\textrm{ess}\,}}}(D_\Gamma )\ge R^*$.

Remark 6.2

(Symmetry of the spectrum and essential spectrum) For Example 2 it is immediate from (6.5) that $\sigma (D_\Gamma )$ is symmetric with respect to the origin; if $z\in \sigma (D_\Gamma )$ then $-z\in \sigma (D_\Gamma )$. We conjecture, based on the numerical results for Examples 3 and 4 and similar calculations, that this same symmetry holds in 2D whenever $\Gamma$ is a dilation invariant graph satisfying (4.44). If this conjecture is true, then, for all $\Gamma \in \mathscr {D}_\mathcal {A}$, $\sigma _{{{\,\textrm{ess}\,}}}(D_\Gamma )$ is symmetric with respect to the origin by Theorem 6.1. We note that symmetry results in the 2D case of this sort are proved for $D_\Gamma :L^2(\Gamma )\rightarrow L^2(\Gamma )$ when $\Gamma$ is the boundary of a bounded $C^2$ domain $\Omega _-$ in [35, Proposition 6], and for $D^\prime _\Gamma$ as an operator on the natural energy space for general Lipschitz $\Omega _-$ [25, Theorem 2.1] (in both these cases the relevant spectrum lies in $[-1,1]$). In 3D, even when $\Gamma$ is a polyhedron, the spectrum and essential spectrum of $D_\Gamma$ need not be symmetric with respect to the origin (see, e.g., [15]).

Fig. 7

Numerical results for Example 4; $f_+=f$ given by (6.2), $f_- = 0$, $\alpha = \frac{2}{3}$