Flux and symmetry effects on quantum tunneling

Abstract

Motivated by the analysis of the tunneling effect for the magnetic Laplacian, we introduce an abstract framework for the spectral reduction of a self-adjoint operator to a hermitian matrix. We illustrate this framework by three applications, firstly the electro-magnetic Laplacian with constant magnetic field and three equidistant potential wells, secondly a pure constant magnetic field and Neumann boundary condition in a smoothed triangle, and thirdly a magnetic step where the discontinuity line is a smoothed triangle. Flux effects are visible in the three aforementioned settings through the occurrence of eigenvalue crossings. Moreover, in the electro-magnetic Laplacian setting with double well radial potential, we rule out an artificial condition on the distance of the wells and extend the range of validity for the tunneling approximation recently established in Fefferman et al. (SIAM J Math Anal 54: 1105–1130, 2022), Helffer & Kachmar (Pure Appl Anal, 2024), thereby settling the problem of electro-magnetic tunneling under constant magnetic field and a sum of translated radial electric potentials.

1 Introduction

1.1 Tunneling and flux effects

The tunneling induced by symmetries is an interesting phenomenon in spectral theory featuring an exponentially small splitting between the ground state and the next excited state energies. The magnetic flux has an effect on the eigenvalue multiplicity which can lead to oscillatory patterns in the spectrum: as the magnetic flux varies, the eigenvalues may cross and split infinitely many times. Hence it is interesting to look at the interaction between symmetry and flux effects. We explore this question by investigating examples of operators involving the magnetic Laplacian $(-\mathrm ih\nabla -{\textbf{A}})^2$ perturbed in various ways by an electric potential, a boundary condition or a magnetic field discontinuity. We observe interesting flux effects, manifested in endless eigenvalue crossings, when adding symmetry assumptions on the electric potential, the boundary of the domain or the magnetic field discontinuity set.

A braid structure in the distribution of the low lying eigenvalues was predicted heuristically [11, Sec. 15.2.4] and confirmed numerically [4] for the magnetic Laplacian on an equilateral triangle with Neumann boundary condition and constant magnetic field. We confirm this prediction by giving a proof for smoothed equilateral triangles and for other examples on the full plane (electro-magnetic Laplacian and magnetic steps).

Let us introduce a mathematical definition of (semi-classical) braid structure of lowest eigenvalues. Consider a family of unbounded self-adjoint operators $(T_h)_{h\in (0,1]}$ on a Hilbert space H. Let us assume that, for every $h\in (0,1]$, $T_h$ is semi-bounded from below and denote by $\lambda _1(T_h),\lambda _2(T_h),\cdots$ the discrete eigenvalues below the essential spectrum of $T_h$, counted with multiplicity. In practical examples, the parameter h will be the semi-classical parameter, which tends to 0 and can result as the inverse of the magnetic field’s intensity in problems involving strong magnetic fields.

Definition 1.1

The lowest eigenvalues of $T_h$, $\lambda _1(T_h)$ and $\lambda _2(T_h)$, are said to have a braid structure, if

$$\begin{aligned} \forall \,\epsilon>0, ~\exists \,h_\epsilon ,h'_\epsilon \in (0,\epsilon ),\ {\left\{ \begin{array}{ll} \lambda _2(T_{h_\epsilon })-\lambda _1(T_{h_\epsilon })>0\\ \lambda _2(T_{h'_\epsilon })-\lambda _1(T_{h'_\epsilon })=0 \end{array}\right. }. \end{aligned}$$

According to Definition 1.1, the eigenvalues $\lambda _1(T_h)$ and $\lambda _2(T_h)$ exhibit infinitely many crossings and splittings as the parameter h varies in a right neighborhood of 0 (see Fig. 1). This phenomenon has also been observed in non-simply connected domains when considering a magnetic Laplacian and assuming an Aharonov–Bohm magnetic potential. Furthermore, it has been proven in [23] that these crossings and splittings occur in response to variations in the magnetic flux.

Fig. 1

A schematic figure of eigenvalues with a braid structure, occurring in the presence of trilateral symmetry. The ground state energy has multiplicity 2 infinitely many times. Observe also that the energy of the second state may have multiplicity 2 while the ground state energy is a simple eigenvalue

For the semi-classical magnetic Laplacian on a simply connected domain with Neumann boundary conditions, the spectrum is related with the spectral properties of an operator which is defined on the boundary. Hence we actually work on another non-simply connected domain (i.e. the boundary) and therefore flux effects are expected to exhibit crossings and splittings of eigenvalues. However, this is not the case when the boundary curvature has for example a unique non-degenerate maximum. In this case the eigenvalues split in the semi-classical limit [12]. It is when non-degenerate minima are exchanged in the presence of symmetries (like in the case of an ellipse or a smoothed equilateral triangle), that eigenvalue crossings are expected to occur along with tunneling effects [6, 11]. We will also prove such a behavior for the electro-magnetic Laplacian with ‘potential’ wells located on the vertices of an equilateral triangle, which to our knowledge is quite novel.

1.2 Electro-magnetic tunneling

The analysis in this paper yields new results on the electro-magnetic Laplacian on ${\mathbb {R}}^2$,

$$\begin{aligned} {\mathcal {L}}_{h,b} = (-\mathrm ih\nabla -b{\textbf{A}})^2+V, \end{aligned}$$

where b, h are positive parameters, ${\textbf{A}}=\frac{1}{2}(-x_2,x_1)$ is the vector field generating the unit uniform magnetic field, ${{\,\textrm{curl}\,}}{\textbf{A}}=1$, and V is a smooth function.

What we call the wells are the points where V attains its minimum. The pure electric case where $b=0$ was settled for any number of wells n in [21]. We would like to address the case where $b>0$ and $n\ge 2$. For $n=2$, this problem was considered in [22, 25] and revisited recently in [10, 17]. The article [22] follows a perturbative approach (i.e. considers the case b relatively small) and assumes the analyticity of the electric potential V, while the results in [10, 17] hold for any $b>0$ but under the assumption that V is non-positive and defined as a superposition of radially symmetric compactly supported functions.

1.2.1 Double wells

Suppose that the electric potential V is as follows

$$\begin{aligned} V(x)=v_0(|x-z_1|)+v_0(|x-z_2|) \end{aligned}$$

(1.1)

where $v_0\in C^\infty (\overline{{\mathbb {R}}_+})$ vanishes on $[a,+\infty )$, negative-valued on [0, a) and has a unique and non-degenerate minimum at 0. The wells of V are then $z_1$ and $z_2$. We prove the following theorem that, in particular, rules out eigenvalue crossings for double wells.

Theorem 1.2

Assuming $b>0$ is fixed and V is given as in (1.1) with $z_1=0$, $z_2=(L,0)$ and $L\ge 4a$, then there exists a positive constant ${\mathscr {E}}_{b,L}(v_0)$ such that

$$\begin{aligned} h \ln \big (\lambda _2({\mathcal {L}}_{h,b})-\lambda _1({\mathcal {L}}_{h,b})\big ) \underset{h\searrow 0}{\sim }\ - {\mathscr {E}}_{b,L}(v_0). \end{aligned}$$

(1.2)

Moreover

$$\begin{aligned} {\mathscr {E}}_{b,L}(v_0)\underset{b\searrow 0}{\sim }2\int _0^{L/2}\sqrt{v_0(\rho )-v_0^{\min }}\mathop {}\!{\textrm{d}}\rho , \end{aligned}$$

where $v_0^{\min }=\min _{r\ge 0}v_0(r)$.

Remark 1.3

• i) The asymptotics in (1.2) was obtained earlier in [17] for $b=1$ but under the assumption that

  $$\begin{aligned} L > 4\Bigl (\sqrt{|v_0^{\textrm{min}}|}+a\Bigr ). \end{aligned}$$

  (1.3)

• ii) The use of (1.3) in [17] was technical. In fact, assuming (1.3), it is proved in [10] that

  $$\begin{aligned} \lambda _2({\mathcal {L}}_{h,1})-\lambda _1({\mathcal {L}}_{h,1})\underset{h\searrow 0}{\sim } {\mathfrak {c}}_h( v_0,L) \end{aligned}$$

  (1.4)

  where ${\mathfrak {c}}_h(v_0,L)$ is the hopping coefficient that will be introduced in (3.15) later on. The accurate approximation of $\ln {\mathfrak {c}}_h(v_0,L)$ was then carried out in [17].

• iii) For $b\not =1$, by a change of semi-classical parameter, the condition in (1.3) reads as follows

  $$\begin{aligned} L>4\Bigl (b^{-1}\sqrt{|v_0^{\textrm{min}}|} + a\Bigr ). \end{aligned}$$

  Clearly, this is a very strong condition on L which in particular prevents us from considering the limit $b\searrow 0$. The novelty in Theorem 1.2 is in improving the previous condition which could appear as artificial. However, it is still an open question whether (1.2) holds for $2a<L<4a$. We provide a sufficient condition on $(L,a,v_0)$ that allows for L to be slightly below 4a; see Assumption 3.6.

• iv) The dependence on b in the expression of ${\mathscr {E}}_{b,L}(v_0)$ can actually be made more explicit. We have indeed

  $$\begin{aligned} {\mathscr {E}}_{b,L}(v_0)= b\, S(b^{-2}v_0,L)\,, \end{aligned}$$

  where $S(\cdot ,L)$ will be introduced in (3.18) later on.

  The leading term of ${\mathscr {E}}_{b,L}(v_0)$ in the limit $b\searrow 0$ was calculated in [17, Rem. 1.3], and it is consistent with the existing results [20, 29] without a magnetic field, $b=0$, thereby showing a sort of continuity of the tunneling estimate with respect to the magnetic field’s strength. Studying the transition from $b=0$ to $b>0$ when $b=b(h)$ could be an interesting question.

1.2.2 Three wells and eigenvalue crossings

Suppose now that the electric potential V is as follows

$$\begin{aligned} V(x)=v_0(|x-z_1|)+v_0(|x-z_2|)+v_0(|x-z_3|) \end{aligned}$$

(1.5)

where $v_0$ is the same function as in (1.5) and that the wells $z_1,z_2,z_3$ are located on the vertices of an equilateral triangle with side length L. We prove then the existence of a braid structure in the sense of Definition 1.1.

Theorem 1.4

Assuming $b>0$ is fixed and V is given as in (1.5) with

$$\begin{aligned} |z_1-z_2|=|z_2-z_3|=|z_3-z_1|=L>4a, \end{aligned}$$

then the lowest egigenvalues of ${\mathcal {L}}_{h,b}$ has a braid structure. Moreover,

$$\begin{aligned} \limsup _{h\searrow 0} \Bigl (h \ln \big (\lambda _2({\mathcal {L}}_{h,b})-\lambda _1({\mathcal {L}}_{h,b})\big )\Bigr ) = - {\mathscr {E}}_{b,L}(v_0) \end{aligned}$$

with ${\mathscr {E}}_{b,L}(v_0)$ the same positive quantity as in Theorem 1.2.

Not only Theorem 1.4 establishes the existence of infinitely many eigenvalue crossings and splittings, but it also establishes an accurate estimate for the magnetic tunneling induced by three symmetric potential wells, thereby extending the recent results of [10, 17] on double wells.

1.3 Geometrically induced braid structure

We present here results on the pure magnetic Laplacian where the eigenvalue crossings are induced by a combination of the geometry and the flux in the semi-classical limit. We shall describe the results when $\Omega$ is a smoothed triangle (see Fig 2). That is, $\Omega$ is a simply connected domain, invariant under rotation by $2\pi /3$, with three points of maximum curvature that are equidistant with respect to the arc-length distance on the boundary.

Fig. 2

Illustration of the domain $\Omega$: a smoothed triangle invariant under the rotation by $2\pi /3$

1.3.1 The magnetic Neumann Laplacian under constant magnetic field

In the Hilbert space $L^2(\Omega )$, we consider the magnetic Neumann Laplacian ${\mathscr {L}}_h^N=(-\mathrm ih\nabla -{\textbf{A}})^2$ on $\Omega$, with uniform magnetic field ${{\,\textrm{curl}\,}}{\textbf{A}}=1$ and (magnetic) Neumann condition

$$\begin{aligned} \nu \cdot (-\mathrm ih\nabla -{\textbf{A}})u|_{\partial \Omega }=0. \end{aligned}$$

Theorem 1.5

Let the domain $\Omega$ be a smoothed triangle, invariant under the rotation by $2\pi /3$ (see Fig. 2). The lowest eigenvalues of the operator ${\mathscr {L}}_h^N$ have a braid structure.

The presence of the Neumann boundary condition plays a vital role in the preceding theorem. This condition is responsible for the semi-classical localization of the bound states near the boundary of $\Omega$ and this is this localization that gives rise to the observed flux effects.

1.3.2 The Landau Hamiltonian under a magnetic step

We consider here the Landau Hamiltonian ${\mathscr {L}}_h^B=(-\mathrm ih\nabla -{\textbf{A}})^2$ on ${\mathbb {R}}^2$ with the discontinuous magnetic field

$$\begin{aligned} B ={{\,\textrm{curl}\,}}{\textbf{A}}= {\left\{ \begin{array}{ll} 1 &{}\text { on }\Omega ,\\ \vartheta \in (-1,0)&{}\text { on }{\mathbb {R}}^2\setminus {\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

Such magnetic fields have been called ‘magnetic steps’ in the literature, and the semi-classical limit for the operator ${\mathcal {L}}_h^B$ has been studied recently in [3, 13] (and in [14] for $\vartheta =-1$). The bound states of the system become increasingly concentrated along the discontinuity of B in the semi-classical limit. Consequently, we expect the emergence of flux effects, and they are indeed realized when $\Omega$ is a smoothed triangle with 3-fold symmetry.

Theorem 1.6

Assume that $\Omega$ is a smoothed triangle, invariant under the rotation by $2\pi /3$ (see Fig. 2). The lowest eigenvalues of the operator ${\mathscr {L}}_h^B$ have in the semi-classical limit a braid structure.

We therefore have an example where the eigenvalues of the Landau Hamiltonian on the full plane cross and split infinitely many times. This is the consequence of having a sign changing magnetic field with a discontinuity along a simple smooth curve.

Recently, an estimate for the tunneling induced by a smooth magnetic field that vanishes non-degenerately along a smooth curve has been established in [1]. It seems natural to expect the existence of a braid structure in that setting too when adding symmetry assumptions.

1.4 Organization

The paper is organized as follows. Since we work with various symmetry configurations, Section 2 is devoted to an abstract spectral reduction to a hermitian matrix (so often called the interaction matrix in the literature on tunneling effects [16, 20]). This provides us with a robust methodology when analyzing tunneling effects in various settings. Loosely speaking, all we need is the construction of adequate quasi-modes.

In Sect. 3 we discuss the electro-magnetic Laplacian. Our investigation has two ingredients, the first is to appply the abstract methodology in Sect. 2, and the second is to control the errors produced by the interaction terms; the later task is achieved by using the analysis in the recent work [17]. Section 3 concludes with the proofs of Theorems 1.2 and 1.4.

In Sect. 4, we prove Theorem 1.5 by applying the results of Sect. 2. The control of the error terms and the computation of the interaction term were done in [6].

Finally, in Sect. 5, we prove Theorem 1.6 by applying the constructions in Sect. 2. We will be more succinct here since the analysis is similar to Sect. 4.1.2. The control of the error terms and the asymptotics of the interaction terms were done in [13] and are actually rather close to those in [6].

2 Abstract framework for a spectral reduction to a hermitian matrix

The strategy of reducing the spectral analysis of Schrödinger operators to that of a hermitian matrix goes back to [20]. It was presented in the survey works [9, 16], and was adapted to other type of operators [6, 18, 19]. Nevertheless, it should be emphasized that the scheme of proof (as described for example in [9]) can not be immediately transposed to some of the cases considered in our applications, due to the absence of optimal Agmon estimates.

In this section we consider a family of operators dependent on a positive semi-classical parameter $h\ll 1$. Assuming the existence of certain quasi-modes (see Assumption 2.1 below), we can approximate the eigenvalues of the operator with those of the matrix of its restriction on a specific basis (Proposition 2.4 below). Assuming an additional symmetry hypothesis (Assumption 2.6 below) we can relabel the eigenvalues and spot their possible crossings and splittings as the semi-classical parameter approaches 0 (Eqs. (2.20), (2.21) and Paragraphs 2.4.2, 2.4.3).

2.1 Preliminaries

Consider a Hilbert space H endowed with an inner product $\langle \cdot ,\cdot \rangle$ and a family of self-adjoint unbounded operators $T_h:D_h\rightarrow H$, $h\in (0,1]$. Assume furthermore that, for every $h\in (0,1]$, $T_h$ is semi-bounded from below and has a sequence of discrete eigenvalues

$$\begin{aligned} \lambda _1(h) \le \lambda _2(h) \le \lambda _3(h) \le \ldots < \Sigma _h :=\inf \sigma _{\textrm{ess}}(T_h) \in {\mathbb {R}}\cup \{+\infty \}, \end{aligned}$$

counted with multiplicity. By the min-max principle the eigenvalues can be represented as

$$\begin{aligned} \lambda _n(h) = \inf _ {\begin{array}{c} M\subset D_h\\ \dim (M)=n \end{array}} \left( \max _{\begin{array}{c} \phi \in M\\ \Vert \phi \Vert =1 \end{array}} \langle T_h\phi ,\phi \rangle \right) . \end{aligned}$$

We will work under additional assumptions on the operators $(T_h)_{h\in (0,1]}$.

Assumption 2.1

(n wells) Let $n\ge 2$ be an integer. There exist positive constants ${\mathfrak {S}}_1,{\mathfrak {S}}_2,{\mathfrak {S}}_3, c,p$ and $h_0 \in (0,1]$ such that, for all $h\in (0,h_0]$, there exists a subspace $E_h= {{\,\textrm{span}\,}}(u_{h,1},\ldots ,u_{h,n})\subset D_h$ such that:

• (1) $\max _{1\le i \le n} \Vert T_h u_{h,i} \Vert ={\mathcal {O}}(\mathrm e^{-{\mathfrak {S}}_1/h}).$

• (2) $\langle u_{h,i},u_{h,j} \rangle = {\left\{ \begin{array}{ll} 1+{\mathcal {O}}(\mathrm e^{-{\mathfrak {S}}_2/h}) &{} i = j, \\ {\mathcal {O}}(\mathrm e^{-{\mathfrak {S}}_3/h}) &{} i \ne j. \end{array}\right. }$

• (3) $\lambda _{n+1}(h) \ge c\,h^{p}$.

It results from (2) above that $\dim (E_h)=n$ for h small enough. As a consequence of Assumption 2.1, we now prove that the operator $T_h$ has precisely n eigenvalues that are exponentially small in h, and there is a gap to $\lambda _{n+1}(h)$, since it is at least of polynomial size in h.

Proposition 2.2

Under Assumption 2.1, there exist positive constants $C,h_1$ such that, for $h\in (0,h_1]$,

$$\begin{aligned} \lambda _n(h)\le C e^{-\mathfrak {S}_1/h}. \end{aligned}$$

(2.1)

In particular $\lambda _n(h)<\lambda _{n+1}(h)$ for h sufficiently small.

Proof

Since $\dim (E_h)=n$, we can use the min-max principle. Let $\phi = \sum _{j}\alpha _ju_{h,j}$ be in $E_h$. Then, the triangle inequality and Cauchy–Schwarz inequality, together with (1) and (2) from Assumption 2.1, provide the existence of constants $A_1$ and $A_2$, such that if h is small enough,

$$\begin{aligned} \langle T_h\phi ,\phi \rangle \le \sum _{j,k} |\alpha _j|\cdot |\alpha _k| \cdot \Vert T_h u_{h,j} \Vert \cdot \Vert u_{h,k}\Vert \le A_1 \mathrm e^{-\mathfrak {S}_1/h}(1+A_2\mathrm e^{-\mathfrak {S}_2/h}) \sum _{j}|\alpha _j|^2. \end{aligned}$$

On the other hand, we can use (2) of Assumption 2.1 to bound the norm of $\phi$ from below. We get indeed constants $A_3$ and $A_4$ such that, if h is small enough,

$$\begin{aligned} \Vert \phi \Vert ^2 \ge (1- A_3\mathrm e^{-\mathfrak {S}_2/h} - A_4\mathrm e^{-\mathfrak {S}_3/h}) \sum _j|\alpha _j|^2. \end{aligned}$$

This gives the bound in (2.1). Combining this bound with (3) in Assumption 2.1 we conclude that that $\lambda _{n+1}(h) > \lambda _n(h)$ if h is sufficiently small. $\square$

We want to link the quasi mode constructions $\{u_{h,j}\}$ to the low-lying eigenvalues of $T_h$. To do this, we want to show that the symmetric matrix $\textsf{U}_h=(\textsf{u}_{j,k})$,

$$\begin{aligned} \textsf{u}_{j,k} = \langle T_h u_{h,j},u_{h,k}\rangle , \end{aligned}$$

(2.2)

does not differ much (component-wise) from the matrix $\textsf{W}_h$ that will be constructed as the restriction of $T_h$ to the eigenspace

$$\begin{aligned} F_h:=\bigoplus _{j=1}^n {{\,\textrm{Ker}\,}}(T_h-\lambda _j(h)), \end{aligned}$$

(2.3)

written in an orthonormal basis. We will do the approximation in two steps. We first consider the projected functions

$$\begin{aligned} v_{h,j} = \Pi _{F_h}u_{h,j} \end{aligned}$$

and show that the norms $\Vert v_{h,j} - u_{h,j}\Vert$ are small. Since the span of $\{u_{h,j}\}$ is n-dimensional by Assumption 2.1 (2), it will follow that the $\{v_{h,j}\}$ are linearly independent, and thus constitute a basis for $F_h$.

Proposition 2.3

If Assumption 2.1 holds for $E_h$, then for $h>0$ sufficiently small, we have $\dim (F_h)=n$, and the vectors

$$\begin{aligned} v_{h,i}=\Pi _{F_h}u_{h,i}\quad (i\in \{1,\cdots ,n\}), \end{aligned}$$

form a basis of $F_h$. Moreover they satisfy

$$\begin{aligned} \max _{1\le i\le n} \Vert v_{h,i}-u_{h,i}\Vert = {\mathcal {O}}(h^{-p}\mathrm e^{-{\mathfrak {S}}_1/h}). \end{aligned}$$

(2.4)

Proof

Since we count multiplicities, we know in general that $\dim (F_h)\ge n$. However, by (3) in Assumption 2.1, we get from Proposition 2.2 that $\dim (F_h)=n$.

With $v_{h,i}=\Pi _{F_h} u_{h,i}$ we note that $u_{h,i} - v_{h,i} \in H \ominus F_h$. Since $T_h$, restricted to $H \ominus F_h$ is bounded below by $ch^p$ by Assumption 2.1 (3), we find that

$$\begin{aligned} \Vert T(u_{h,i} - v_{h,i})\Vert \ge c\, h^p\Vert u_{h,i}-v_{h,i}\Vert . \end{aligned}$$

On the other hand, According to Assumption 2.1 (1) and (2), and Proposition 2.2,

$$\begin{aligned} \Vert T(u_{h,i} - v_{h,i})\Vert \le \Vert Tu_{h,i}\Vert + \Vert T v_{h,i}\Vert \le C\mathrm e^{-\mathfrak {S}_1/h}. \end{aligned}$$

Combining these inequalities we get (2.4). From this and Assumption 2.1 (2), we find that $\{v_{h,1},\ldots ,v_{h,n}\}$ are linearly independent, and hence a basis for $F_h$. $\square$

2.2 Reduction to a matrix through a suitable orthonormal basis

The aim in this subsection is to find an orthonormal basis for $F_h$ such that the matrix of the restriction of $T_h$ in this basis can be well approximated. Later in the applications to multiple wells problems this matrix will be according to the previous literature called the interaction matrix.

The basis $\{v_{h,j}\}$ of $F_h$ that we just constructed will, in general, not be orthogonal. We construct, by a symmetry-preserving Gram–Schmidt procedure an orthonormal basis $\{w_{h,j}\}$. The matrix $\textsf{W}_h$ will be the matrix of $T_h$ restricted to $F_h$, written in this new basis $\{w_{h,j}\}$.

Let us denote by ${\textsf{G}}_h=({\textsf{g}}_{ij}(h))_{1\le i,j\le n}$ the Gram matrix of the basis $\{v_{h,1},\ldots ,v_{h,n}\}$ of $F_h$, where

$$\begin{aligned} {\textsf{g}}_{ij}=\langle v_{h,i},v_{h,j}\rangle . \end{aligned}$$

(2.5)

Since the $\{v_{h,j}\}$ are linearly independent, the Gram matrix becomes positive definite, so ${\textsf{G}}_h^{-1/2}$ is well defined and positive definite. We obtain an orthonormal basis ${\mathcal {V}}_h=\{w_{h,1},\ldots ,w_{h,n}\}$ of $F_h$ as follows¹

$$\begin{aligned} \begin{pmatrix} w_{h,1}\\ \vdots \\ w_{h,n} \end{pmatrix} = {\textsf{G}}_h^{-1/2} \begin{pmatrix} v_{h,1}\\ \vdots \\ v_{h,n} \end{pmatrix}. \end{aligned}$$

(2.6)

We consider the restriction of $T_h$ to the space $F_h$ and denote by ${\textsf{W}}_h=({\textsf{w}}_{ij})_{1\le i,j\le n}$ its matrix in the basis ${\mathcal {V}}_h$, so $\textsf{w}_{ij}=\langle T_h w_{h,i},w_{h,j}\rangle$. The matrix ${\textsf{W}}_h$ is hermitian, with eigenvalues $\{\lambda _1(h),\ldots ,\lambda _n(h)\}$. The next proposition controls how ${\textsf{W}}_h$ is approximated by the matrix ${\textsf{U}}_h$ defined by (2.2).

Proposition 2.4

Let

$$\begin{aligned} \left. \begin{aligned} \Lambda _h&:=\Vert T_h|_{F_h}\Vert =\max _{1\le i\le n}|\lambda _i(h)|,\\ C_h&= h^{-p}\mathrm e^{-{\mathfrak {S}}_1/h}+\mathrm e^{-\min ({\mathfrak {S}}_2,{\mathfrak {S}}_3)/h},\\ \varepsilon _h&=(\Lambda _h+C_h)C_h . \end{aligned} \right\} \end{aligned}$$

(2.7)

Then the matrix ${\textsf{R}}_h = (\textsf{r}_{ij})$,

$$\begin{aligned} {\textsf{R}}_h :=\textsf{W}_h - {\textsf{U}}_h, \end{aligned}$$

(2.8)

is symmetric, and satisfies

$$\begin{aligned} {\textsf{r}}_{ij} = {\mathcal {O}}(\varepsilon _h)\,. \end{aligned}$$

(2.9)

Proof

Step 1. Proposition 2.3 says that

$$\begin{aligned} \langle v_{h,i},v_{h,j}\rangle =\langle u_{h,i},u_{h,j}\rangle +{\mathcal {O}}(h^{-p}\mathrm e^{-{\mathfrak {S}}_1/h}). \end{aligned}$$

With ${\textsf{I}}$ denoting the $n\times n$ identity matrix, we get from (2) in Assumption 2.1,

$$\begin{aligned} {\textsf{G}}_h={\textsf{I}}+{\mathcal {O}}(C_h) \end{aligned}$$

so that

$$\begin{aligned} {\textsf{G}}_h^{-1/2}={\textsf{I}}+{\mathcal {O}}(C_h). \end{aligned}$$

It follows then that

$$\begin{aligned} \left. \begin{aligned} \Vert w_{h,i}-v_{h,i}\Vert&={\mathcal {O}}(C_h),\\ \Vert T_h w_{h,i}-T_h v_{h,i}\Vert&={\mathcal {O}}\big (\Lambda _hC_h\big ) \end{aligned} \right\} \end{aligned}$$

(2.10)

where in the second estimate in (2.10), we simply combine the first estimate and the definition of $\Lambda _h$.

Step 2. We may write

$$\begin{aligned} {\textsf{w}}_{ij}{} & {} = \langle T_{h}w_{h,i},w_{h,j}\rangle = \langle T_hv_{h,i},v_{h,j}\rangle \\{} & {} \quad + \langle T_hv_{h,i},w_{h,j}-v_{h,j}\rangle + \langle T_h (w_{h,i}-v_{h,i}),w_{h,j}\rangle . \end{aligned}$$

Using this identity, (2.10) and Proposition 2.3, we get

$$\begin{aligned} \langle T_h w_{h,i},w_{h,j}\rangle = \langle T_h v_{h,i},v_{h,j}\rangle + {\mathcal {O}}(\Lambda _hC_h). \end{aligned}$$

(2.11)

Then, we use

$$\begin{aligned} \langle T_hv_{h,i},v_{h,j}\rangle = \langle T_h u_{h,i},u_{h,j}\rangle + \langle T_h u_{h,i},v_{h,j}-u_{h,j}\rangle + \langle v_{h,i}-u_{h,i}, T_hv_{h,j}\rangle . \end{aligned}$$

By Proposition 2.3 and (1) in Assumption 2.1, we have

$$\begin{aligned} \begin{aligned} \langle T_hu_{h,i},v_{h,j}-u_{h,j}\rangle&= {\mathcal {O}}(h^{-p}\mathrm e^{-2{\mathfrak {S}}_1/h})={\mathcal {O}}(C_h^2)\,,\\ \langle v_{h,i}-u_{h,i}, T_hv_{h,j}\rangle&={\mathcal {O}}(\Lambda _hC_h). \end{aligned} \end{aligned}$$

So we get

$$\begin{aligned} \langle T_hv_{h,i},v_{h,j}\rangle = \langle T_h u_{h,i},u_{h,j}\rangle + {\mathcal {O}} (\varepsilon _h)\,, \end{aligned}$$

(2.12)

which together with (2.11) implies (2.9). $\square$

An immediate consequence of Proposition 2.4 is an improved lower bound on the lowest eigenvalue $\lambda _1(h)$.

Corollary 2.5

If Assumption 2.1 holds, then there exist positive constants $C,h_1$ such that, for $h\in (0,h_1]$,

$$\begin{aligned} \lambda _1(h)\ge -C \mathrm e^{-\min (\mathfrak {S}_1,2\mathfrak {S}_2,2\mathfrak {S}_3)/h}. \end{aligned}$$

(2.13)

Proof

It follows from Hölder’s inequality and (1)–(2) in Assumption 2.1 that ${\textsf{u}}_{ij}={\mathcal {O}}(\mathrm e^{-{\mathfrak {S}}_1/h})$, where ${\textsf{u}}_{ij}$ is introduced in (2.2). By Proposition 2.4, we have

$$\begin{aligned} \Lambda _h\le \Vert {\textsf{W}}_h\Vert \le \Vert {\textsf{U}}_h\Vert +\Vert {\textsf{R}}_h\Vert = {\mathcal {O}}( \mathrm e^{-{\mathfrak {S}}_1/h})+{\mathcal {O}}(\Lambda _h C_h+C_h^2) \end{aligned}$$

which yields

$$\begin{aligned} \bigl (1- {\mathcal {O}}(C_h)\bigr ) \Lambda _h \le {\mathcal {O}}( \mathrm e^{-{\mathfrak {S}}_1/h})+ {\mathcal {O}}( C_h^2). \end{aligned}$$

To conclude, we just notice that $C_h=o(1)$. $\square$

As we will see in the next subsection, we can actually say much more about the spectrum of these matrices, when we impose some symmetry condition involving $T_h$ and the choice of the $u_{h,j}$.

2.3 Implementing invariance assumptions

Our task in this subsection is to analyze the case when the matrix ${\textsf{W}}_h$ of $T_h|_{F_h}$ enjoys certain invariance properties. We shall see that this corresponds to what occurs in the case of symmetric wells in the applications, starting from the double well case as mathematically considered by E. Harrell [15] and later extended to the multiple wells case in [20, 21, 28, 29]. Here we mainly follow in a more abstract way [21] and the heuristic presentation given in [11]. We denote by ${\mathbb {Z}}_n$ the cyclic group of order n and by ${\mathfrak {g}} \mapsto \rho ({\mathfrak {g}})$ a faithful unitary representation of ${\mathbb {Z}}_n$ in H. We denote by $a_n$ its generator, so $a_n^n=e$ where e is the identity element of the group.

In addition to the properties in Assumption 2.1, we assume

Assumption 2.6

• (1) The operator $T_h$ commutes with $\rho ({\mathfrak {g}})$ for all ${\mathfrak {g}} \in {\mathbb {Z}}_n$.

• (2) $u_{h,i+1} = \rho ( a_n) u_{h,i}$ for $1\le i\le n-1$.

Remark 2.7

In the applications considered in this article, the Hilbert space will be $H=L^2(\Omega )$ where $\Omega$ is a domain in ${\mathbb {R}}^2$. We first consider the unitary representation $\rho _0$ of ${\mathbb {Z}}_n$ as the group $G_n$ of the n-fold rotations, i.e. the representation such that

$$\begin{aligned} \rho _0 ( a_n):=g_n \end{aligned}$$

is the rotation in ${\mathbb {R}}^2$ by $2\pi /n$ around the origin in ${\mathbb {R}}^2$.

We let the rotation $g_n$ act on functions as

$$\begin{aligned} \bigl (M(g_n)u\bigr )(x)= u ( g_n^{-1} x). \end{aligned}$$

(2.14)

This gives by extension to any element of $G_n$ a representation of $G_n$ in $L^2(\Omega )$ if $\Omega \subset {\mathbb {R}}^2$ is a domain invariant by $G_n$ and we then define $\rho$ by

$$\begin{aligned} \rho ({\mathfrak {g}}) = M (\rho _0({\mathfrak {g}} )) \,. \end{aligned}$$

Equivalently to Assumption 2.6, we can then write in this case

Assumption 2.8

• (1) $\Omega \subset {\mathbb {R}}^2$ is a domain invariant by $G_n$ and $H=L^2(\Omega )$.

• (2) The operator $T_h$ commutes with $M(g_n)$.

• (3) $u_{h,i+1} = M( g_n) u_{h,i}=u_{h,1}(g_n^{-i}x)$ for $1\le i\le n-1$.

Assumption 2.6 permits to treat more general situations which for example occur in the case of manifolds or in the case of higher dimension.

Remark 2.9

If $n=2$ we can define another group of symmetry ${\tilde{G}}$ defined by the reflection . We can then consider a variant of Assumption 2.6 by instead assuming that the domain $\Omega$ is invariant by the reflection ${\tilde{g}}_2$ and that $u_{h,2}=\overline{M({\tilde{g}}_2)u_{h,1}}$. This symmetry invariance was assumed by the papers considering the magnetic tunneling induced by the geometry of the domain [1, 6, 13, 24]. Notice that $M ({\tilde{g}}_2)$ does not commute with $T_h$ and that we have consequently to compose $M({\tilde{g}}_2)$ with the complex conjugation $\Gamma$ in order to get

$$\begin{aligned} T_h \Gamma M({\tilde{g}}_2) = \Gamma M({\tilde{g}}_2) T_h \,. \end{aligned}$$

Proposition 2.10

If Assumption 2.6 holds, then the orthonormal basis ${\mathcal {V}}_h=\{w_{h,1},\ldots ,w_{h,n}\}$ of $F_h$ satisfies,

$$\begin{aligned} w_{h,i+1} = \rho ( a_n) w_{h,i} \quad (1\le i\le n-1). \end{aligned}$$

Proof

Recall that $\{w_{h,1},\ldots ,w_{h,n}\}$ is defined in (2.6) by the Gram matrix starting from the basis consisting of the vectors $v_{h,i}=\Pi _{F_h}u_{h,i}$, $1\le i\le n$. It suffices to observe that the projector $\Pi _{F_h}$ on the eigenspace $F_h$ commutes with $\rho (a_n)$. Actually, since $T_h$ commutes with $\rho (a_n)=M(g_n)$, we get, for every z in the resolvent set of $T_h$, that $(z-T_h)^{-1}\rho (a_n)=\rho (a_n)(z-T_h)^{-1}$, and consequently, the identity $\Pi _{F_h}\rho (a_n)=\rho (a_n)\Pi _{F_h}$ follows from Cauchy’s formula.

Let us give a more hands-on argument for this commutativity. If $\{w_{h,1},\ldots ,w_{h,n}\}$ is an orthonormal basis of $F_h$ consisting of eigenvectors of $T_h$, then, since $T_h$ commutes with $\rho ( a_n)$, we get that $\rho ( a_n)w_{h,1},\ldots ,\rho ( a_n)w_{h,n}$ are eigenvectors of $T_h$ and form an orthonormal basis of $F_h$. Consequently

$$\begin{aligned} \begin{aligned} \Pi _{F_h} \rho ( a_n)u&= \sum _{k=1}^n\langle \rho ( a_n)u,\rho ( a_n)w_{h,k}\rangle \rho ( a_n)w_{h,k}\\&= \sum _{k=1}^n\langle u,w_{h,k}\rangle \rho ( a_n)w_{h,k}=\rho ( a_n)\Pi _{F_h}u. \end{aligned} \end{aligned}$$

$\square$

The matrix of $\rho ( a_n)$ in the basis ${\mathcal {V}}_h$ is the same as the matrix of the shift operator $\tau$ on $\ell ^2({\mathbb {Z}}/n{\mathbb {Z}})$, whose matrix is given by

$$\begin{aligned} \tau _{j,k} = \delta _{j+1,k} \text{ for } 1\le j,k\le n \end{aligned}$$

(2.15)

where $\delta _{i,k}$ denotes the Kronecker symbol, with i computed in ${\mathbb {Z}}/n{\mathbb {Z}}$. When $n=2$ and $n=3$, the matrix $\tau$ is respectively given by

$$\begin{aligned} \begin{pmatrix} 0 &{}\quad 1 \\ 1 &{}\quad 0 \end{pmatrix} \quad \text {and}\quad \begin{pmatrix} 0 &{}\quad 0 &{}\quad 1 \\ 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 \end{pmatrix}\,. \end{aligned}$$

We observe that

$$\begin{aligned} \tau ^{n-1}=\tau ^{-1}=\tau ^*\,. \end{aligned}$$

The property that the operator $T_h$ commutes with $\rho ( a)$ implies that the matrix ${\textsf{W}}_h$ (of $T_h|_{F_h}$ in the basis ${\mathcal {V}}_h$) commutes with $\tau$, i.e., $\tau {\textsf{W}}_h = {\textsf{W}}_h \tau$. Note that this invariance condition yields that

$$\begin{aligned} {\textsf{W}}_h = \sum _{k=0}^{n-1} I_k(h) \tau ^k, \end{aligned}$$

(2.16)

for some coefficients $I_0(h), \ldots , I_{n-1}(h) \in {\mathbb {C}}$. Here $\tau ^0$ denotes the identity matrix.

The Hermitian property of $\textsf{W}_h$ gives, in addition,

$$\begin{aligned} I_0(h) \in {\mathbb {R}}\,,\, I_k(h) = \overline{I_{n-k}(h)}\; \text{ for } k=1,\dots , n-1\,. \end{aligned}$$

(2.17)

Notice that the matrix ${\textsf{U}}_h$ introduced in (2.2) satisfies the same properties as ${\textsf{W}}_h$. Hence we can also write

$$\begin{aligned} {\textsf{U}}_h = \sum _{k=0}^{n-1} J_k(h) \tau ^k \,, \end{aligned}$$

(2.18)

for some coefficients $J_0(h), \ldots ,J_{n-1}(h) \in {\mathbb {C}}\,$ and the Hermitian property of ${\textsf{U}}_h$ also implies

$$\begin{aligned} J_0(h) \in {\mathbb {R}}\,,\, J_k(h) = \overline{J_{n-k}(h)}\; \text{ for } k=1,\dots , n-1\,. \end{aligned}$$

(2.19)

All these invariant matrices (${\textsf{W}}_h$ or ${\textsf{U}}_h$) share the property to be diagonalizable in the same orthonormal basis of eigenfunctions $\mathrm e_k$ $(k=1,\dots ,n)$ whose coordinates in our selected basis are given by

$$\begin{aligned} (\mathrm e_k)_\ell = \omega _n^{(k-1) \ell }\,, \quad \text {with}\quad \omega _n:=\exp (2 i \pi /n)\,. \end{aligned}$$

It is then easy to compute the corresponding eigenvalues.

In particular, we get an explicit representation of the eigenvalues of ${\textsf{W}}_h$ which illustrates when $n=3,4$, the possibility of eigenvalue crossings (i.e. change of multiplicity).

• When $n=2$, ${\textsf{W}}_h$ assumes the form $\begin{pmatrix} I_0&{}\quad I_1\\ I_1&{}\quad I_0 \end{pmatrix}$ with $I_1$ real. This matrix has two eigenvalues

  $$\begin{aligned} \lambda _1 = I_0 - |I_1|\,,\, \lambda _2= I_0 + |I_1|\,. \end{aligned}$$

  (2.20)

• When $n=3$, ${\textsf{W}}_h$ assumes the form

  $$\begin{aligned} \textsf{W}_h= \begin{pmatrix} I_0&{}\quad \overline{I_1}&{}\quad I_1\\ I_1&{}\quad I_0&{}\quad \overline{I_1}\\ \overline{I_1}&{}\quad I_1&{}\quad I_0 \end{pmatrix} \end{aligned}$$

  with $I_1=\rho \mathrm e^{\mathrm i\theta }$, $\rho \ge 0$, $\theta \in [0,2\pi )$. This matrix has three eigenvalues

  $$\begin{aligned} \mu _k = I_0 + 2 \rho \cos \left( \theta +(k-1)\frac{2\pi }{3}\right) ,\quad k\in \{1,2,3\}. \end{aligned}$$

  (2.21)

• When $n=4$, we meet the matrix

  $$\begin{aligned} \textsf{W}_h= \begin{pmatrix} I_0&{}\quad \overline{I_1} &{}\quad I_2&{}\quad I_1 \\ I_1&{}\quad I_0 &{}\quad \overline{I_1}&{}\quad I_2 \\ I_2 &{}\quad I_1 &{}\quad I_0 &{}\quad \overline{I_1} \\ \overline{I_1}&{}\quad I_2&{}\quad I_1 &{}\quad I_0 \end{pmatrix} \end{aligned}$$

  with $I_2$ real and $I_1=\rho \mathrm e^{\mathrm i\theta }$, $\rho \ge 0$, $\theta \in [0,2\pi )$. We refer to [11] for a further discussion of this case. Figure 3 illustrates the braid startucture of the eigenvalues of the matrix ${\textsf{W}}_h$.

Fig. 3

A schematic figure of four eigenvalues with a braid structure in the case of a rotational symmetry of order 4

2.4 Applications

2.4.1 Additional hypothesis

We can strengthen the estimate of $\Lambda _h$ in (2.7) if we assume additionally that²

Assumption 2.11

There exists a positive constant ${\mathfrak {S}}$ such that

$$\begin{aligned} {\mathfrak {S}} < 2\min _{1\le j\le 3} {\mathfrak {S}}_j\,, \end{aligned}$$

(2.22)

$$\begin{aligned} J_{k}(h) \underset{h\searrow 0}{=}{\mathcal {O}}(\mathrm e^{- {\mathfrak {S}}/h}) \quad (k=0,\cdots ,n-1), \end{aligned}$$

(2.23a)

and

$$\begin{aligned} |J_1(h)|\underset{h\searrow 0}{=}\mathrm e^{-({\mathfrak {S}}+o(1))/h}. \end{aligned}$$

(2.23b)

Proposition 2.12

There exist positive constants $C,h_0>0$ such that, if Assumptions 2.1 and 2.11 hold, then for all $h\in (0,h_0]$, the symmetric matrix $\textsf{R}_h=({\textsf{r}}_{ij})$ introduced in (2.8) satisfies

$$\begin{aligned} \Vert {\textsf{R}}_h\Vert \underset{h\searrow 0}{=} {\mathcal {O}}\big (\mathrm e^{-3{\mathfrak {S}}/2h}\big )=o\big (|J_1(h)|\big ). \end{aligned}$$

Proof

This follows by applying Proposition 2.4 (the same argument as in Corollary 2.5). Indeed, we have by the first identity in (2.23),

$$\begin{aligned} \Lambda _h={\mathcal {O}}(\mathrm e^{-{\mathfrak {S}}/h})+{\mathcal {O}}(C_h^2)\,. \end{aligned}$$

(2.24)

Inserting (2.24) into the definition of $\varepsilon _h$ in (2.7), we obtain that

$$\begin{aligned} \varepsilon _h = {\mathcal {O}}(\delta _h) \text{ where } \delta _h=C_h\mathrm e^{-{\mathfrak {S}}/h}+C_h^2 = {\mathcal {O}}(\mathrm e^{- 3{\mathfrak {S}}/2h}). \end{aligned}$$

(2.25)

We can improve the bounds (2.9) of the symmetric matrix $\textsf{R}_h$ into ${\textsf{r}}_{ij} = {\mathcal {O}}(\delta _h)$. To finish the proof, we observe that, by the second identity in (2.23), $\delta _h=o(|J_1(h)|)$. $\square$

2.4.2 The case $n=2$

A first consequence of the previous analysis is a full understanding of the case corresponding to $n=2$ where the symmetry $g_2$ reads $(x_1,x_2) \mapsto (-x_1,-x_2)$. Assuming that Assumptions 2.8 and 2.11 hold, we get from (2.20) and Proposition 2.12

$$\begin{aligned} \lambda _2(h)-\lambda _1(h)=2 |J_1(h) |+ o\big (|J_1(h)|\big ). \end{aligned}$$

(2.26)

2.4.3 The case $n=3$, braid structure of eigenvalues

Suppose that Assumptions 2.8 and 2.11 hold with $n=3$. Let $I_0(h),I_1(h)$ be as in (2.16) and let us write

$$\begin{aligned} I_{1}(h)=\rho (h) \mathrm e^{\mathrm i\theta (h)} \text { where } \rho (h)\ge 0 \text { and } \theta (h)\in [0,2\pi ). \end{aligned}$$

Then, by (2.21), we have a relabeling $\mu _1(h),\mu _2(h)$, and $\mu _3(h)$ of the eigenvalues $\lambda _1(h),\lambda _2(h)$, and $\lambda _3(h)$ of $T_h$ with $\rho =\rho (h)$ and $\theta =\theta (h)$. Moreover,

$$\begin{aligned} I_0(h)= J_0(h) +{\mathcal {O}}(\delta _h),\quad I_1(h) = J_1(h)+{\mathcal {O}}(\delta _h)\sim J_1(h) \end{aligned}$$

(2.27a)

with $\delta _h=o\big (J_1(h)\big )$ defined in (2.25) and

$$\begin{aligned} J_0(h)=\langle T_hu_{h,1},u_{h,1}\rangle ,\quad J_1(h)= \langle T_hu_{h,1},u_{h,2}\rangle . \end{aligned}$$

(2.27b)

Notice that there is possibility for eigenvalue crossings between

• $\mu _1(h)$ and $\mu _2(h)$ if $\theta (h) \in \{ 2\pi /3, 5\pi /3 \}$;

• $\mu _2(h)$ and $\mu _3(h)$ if $\theta (h) \in \{ 0, \pi \}$;

• $\mu _1(h)$ and $\mu _3(h)$ if $\theta (h) \in \{ \pi /3, 4\pi /3 \}$.

The point is then to seek an accurate approximation of $\theta (h)$. Notice that (2.27) yields $I_1(h)\underset{h\searrow 0}{\sim }J_1(h)$. Defining $\theta _1(h)$ by $J_1(h) =\rho _1 (h) \mathrm e^{ i \theta _1 (h)}$, we can approximate $\theta (h)$ by $\theta _1(h)$ modulo $2\pi {\mathbb {Z}}$. This could confirm the predicted braid structure mentioned in [4, 11]. We will consider in detail three models where such a phenomenon holds (see Theorem 3.9, Section 4, and Section 5).

3 Electro-magnetic tunneling

3.1 Introduction

In this section, the Hilbert space is $H=L^2({\mathbb {R}}^2)$, and for given $n\in \mathbb {N}$ we consider $g = g_n$ to be rotation around the origin by $2\pi /n$, and M(g) to be as in (2.14). We are interested in the spectrum of the electro-magnetic Schrödinger operator

$$\begin{aligned} {\mathcal {L}}_{h,b}=(-\mathrm ih\nabla -b{\textbf{A}})^2+V, \end{aligned}$$

where $b,h>0$, and

$$\begin{aligned} {\textbf{A}}(x)=\frac{1}{2}(-x_2, x_1)\,. \end{aligned}$$

(3.1)

Notice that ${\textbf{A}}$ generates the constant magnetic field ${{\,\textrm{curl}\,}}{\textbf{A}}=1$. For the potential V we assume that

$$\begin{aligned}{} & {} V\in C^\infty ({\mathbb {R}}^2,{\mathbb {R}})\,, \end{aligned}$$

(3.2a)

$$\begin{aligned}{} & {} V\le 0\,, \end{aligned}$$

(3.2b)

$$\begin{aligned}{} & {} V \text { is invariant by the rotation } g_n\,. \end{aligned}$$

(3.2c)

Moreover, we assume that

$$\begin{aligned} \text {The minimum of } V \text { is attained at } n \text { non-degenerate minima.} \end{aligned}$$

(3.2d)

Then it results from the invariance property of V that these minima are n equidistant points of ${\mathbb {R}}^2\setminus \{0\}$. We will refer to these points as the wells.

Notice that, when dealing with a fixed $b>0$ we can reduce the analysis to the case where $b=1$ by introducing an effective semi-classical parameter $\hbar =b^{-1}h$ so that

$$\begin{aligned} {\mathcal {L}}_{h,b} = b^2\bigl ((-\mathrm i\hbar \nabla -{\textbf{A}})^2+b^{-2}V\bigr ). \end{aligned}$$

(3.3)

So we will assume henceforth that $b=1$. To relate with the discussion in Sect. 2, our operator $T_h$ is the electro-magnetic Laplacian shifted by a certain scalar $\lambda (h)$.

$$\begin{aligned} T_h :={\mathcal {L}}_h-\lambda (h),\quad {\mathcal {L}}_h = (-\mathrm ih\nabla -{\textbf{A}})^2 + V. \end{aligned}$$

(3.4)

Note that the assumption in (3.2c) implies that $T_h$ commutes with M(g). Hence the condition of invariance of the preceding section holds. The shift constant $\lambda (h)$ in (3.4) will be chosen as the ground state energy of a reference single well operator.

3.2 Single well ground states

Let us first discuss the one well operator.

3.2.1 Preliminary discussion and assumptions

There are various possible approaches to create one well problems in the presence of multiple wells. The approach considered in [20, 21] starts from a general electric potential V and introduce suitable Dirichlet conditions to create an infinite barrier leading to a one well problem. The so-called LCAO³ approach, frequently used in Physics and Atomic Chemistry, and considered in [10, 17], particularly applies when V is a superposition of single well potentials.

As in [10, 17], we consider a radial single well potential. More precisely, we assume in this section that $V={\mathfrak {v}}_0$, where ${\mathfrak {v}}_0\in C_c^\infty ({\mathbb {R}}^2)$ is a non-positive radial function satisfying

$$ \begin{aligned} {\left\{ \begin{array}{ll} {\mathfrak {v}}_0(x)=v_0(|x|) ~ \& ~ v_0^{\min }:=\min \limits _{r\ge 0}v_0(r)<0\,,\\ {{\,\textrm{supp}\,}}{\mathfrak {v}}_0\subset \overline{D(0,a)}:=\{x\in {\mathbb {R}}^2\,:\,|x|\le a\}\,,\\ \mathfrak {v}_0^{-1}(v_0^{\textrm{min}}) = \{0\} \quad \& \quad v_0''(0)>0\,. \end{array}\right. } \end{aligned}$$

(3.5)

We choose $\overline{D(0,a)}$ as the smallest closed disc containing $\textrm{supp}\,{\mathfrak {v}}_0$, i.e.

$$\begin{aligned} a = a(v_0):=\inf \{r>0~:~v_0|_{[r,+\infty )}=0\}. \end{aligned}$$

(3.6)

3.2.2 Preliminary inequalities

As in [17], we will encounter various errors of exponential order which are defined in terms of the function $v_0$ and the constant

$$\begin{aligned} L > 2a(v_0)\,. \end{aligned}$$

(3.7)

With $(v_0,L)$ as above, $a=a(v_0)$ and with

$$\begin{aligned} d(r) = d_{v_0} (r) :=\int _0^{r}\sqrt{\frac{\rho ^2}{4}+v_0(\rho )-v_0^{\min }}\mathop {}\!{\textrm{d}}\rho , \end{aligned}$$

(3.8)

we introduce the four constants (we will often skip the depends on $v_0$ and L)

$$\begin{aligned} \left. \begin{aligned} S_0 = S_0(v_0,L)&:=d (L),\\ S_a =S_a(v_0,L)&:=d (a) + d (L-a),\\ \hat{S}_a ={\hat{S}}_a(v_0,L)&:=d (L-a), \\ {\hat{S}}={\hat{S}}(v_0,L)&:=\inf _{0<r<a}\biggl [ \frac{Lr}{2}+d (r)+d (L-r)\biggr ]. \end{aligned} \right\} \end{aligned}$$

(3.9)

Notice also that, since $v_0$ vanishes on $[a,+\infty )$, we can express the constant $S_a$ as follows

$$\begin{aligned} S_a = 2\int _0^a\sqrt{\frac{r^2}{4}+v_0(r)-v_0^{\min }}\mathop {}\!{\textrm{d}}r+\int _0^{L-a}\sqrt{\frac{r^2}{4}-v_0^{\min }}\mathop {}\!{\textrm{d}}r. \end{aligned}$$

It is important when comparing the errors to compare the three constants introduced in (3.9).

Proposition 3.1

Assume that $v_0$ and L satisfy (3.6) and (3.7). Then we have, with $a=a(v_0)$,

$$\begin{aligned} {\hat{S}}_a(v_0,L)< S_a(v_0,L)< {\hat{S}}(v_0,L) < \min \Bigl ( S_0(v_0,L),\frac{La}{2}+S_a(v_0,L)\Bigr ). \end{aligned}$$

(3.10)

Moreover, if $v_0$ and L satisfy $L\ge 4a(v_0)$, then $2{\hat{S}}_a(v_0,L)>{\hat{S}}(v_0,L)$.

Proof

The inequality ${\hat{S}}_a<S_a$ is obvious. The other inequalities in (3.10) are proved in [17, Prop. 3.3]. So assume that $L>4a$ and let us prove that $2{\hat{S}}_a>{\hat{S}}$. Notice that $S_a={\hat{S}}_a+d(a)$ and ${\hat{S}}<\frac{La}{2}+S_a$, hence

$\square$

3.2.3 The one well approximate eigenfunction

Consider the single well operator

$$\begin{aligned} {\mathcal {L}}_h^{\textrm{sw}} = (-\mathrm ih\nabla -{\textbf{A}})^2+{\mathfrak {v}}_0 \end{aligned}$$

(3.11)

whose ground state energy is

$$\begin{aligned} \lambda (h) = \inf \sigma ({\mathcal {L}}_h). \end{aligned}$$

(3.12)

Since ${\mathfrak {v}}_0\le 0$, ${\mathcal {L}}_h^{\textrm{sw}}$ is smaller (in the sense of comparison of self-adjoint operators) than the Landau Hamiltonian $(-\mathrm ih\nabla -{\textbf{A}})^2$. Hence we have by the min-max principle

$$\begin{aligned} \lambda (h)\le h\,. \end{aligned}$$

(3.13)

In light of the conditions on $v_0$ in (3.5), we know from [17, Thm. 2.1] that $\lambda (h)$ is a simple eigenvalue and that ${\mathcal {L}}_h^{\textrm{sw}}$ has a positive radial ground state satisfying, for any relatively compact domain K of ${\mathbb {R}}^2$,

$$\begin{aligned} {\mathfrak {u}}_h(x)=u_h(|x|),\quad \int _{{\mathbb {R}}^2}|{\mathfrak {u}}_h(x)|^2\mathop {}\!{\textrm{d}}x=1,\quad \Vert e^{d(|x|)/h}{\mathfrak {u}}_h(x) \Vert _{H^2(K)}={\mathcal {O}}(h^{-1/2}), \end{aligned}$$

(3.14)

An important term in the context of tunneling is the hopping coefficient defined in [10, 17] (in the case $n=2$) by

$$\begin{aligned} {\mathfrak {c}}_h(v_0,L) = \int _{D(0,a)} {\mathfrak {v}}_0(x){\mathfrak {u}}_h(x){\mathfrak {u}}_h(x_1+L,x_2)\mathrm e^{\frac{\mathrm iLx_2}{2h}}\mathop {}\!{\textrm{d}}x \end{aligned}$$

(3.15)

where $a=a(v_0)$ (see (3.6)).

As observed in [10], the hopping coefficient is a negative real number and an accurate estimate of it was established recently in [17, Sec. 4 & Eq. (4.32)], which we recall below.⁴

Proposition 3.2

Assume that ${\mathfrak {v}}_0$ and L satisfy (3.5) and (3.7). Then there exists a positive constant $S(v_0,L)$ such that

$$\begin{aligned} \lim _{h\searrow 0}\bigl (h\ln |{\mathfrak {c}}_h(v_0,L)|\bigr )=-S(v_0,L). \end{aligned}$$

(3.16)

Moreover,

$$\begin{aligned} S_a(v_0,L)<S(v_0,L)<{\hat{S}}(v_0,L) \end{aligned}$$

(3.17)

where $S_a(v_0,L)$ and ${\hat{S}}(v_0,L)$ are introduced in (3.9).

We shall use Proposition 3.2 later on in the proof of Proposition 3.5 when dealing with n potential wells ($n\ge 2$). Let us recall the definition of $S(v_0,L)$ given in [17, Eq. (4.25) and (4.26)]. We have

$$\begin{aligned} S(v_0,L) :=-F({\mathfrak {v}}_0)+\inf _{\begin{array}{c} r\in [0,a]\\ t\in (0,+\infty ) \end{array}}\Psi (r,t), \end{aligned}$$

(3.18)

where $a=a(v_0)$ is introduced in (3.6) and, with d introduced in (3.8),

$$\begin{aligned} \left. \begin{aligned} F(v_0)&:=\frac{a}{4}\sqrt{a^2+4|v_0^{\min }|} + \frac{|v_0^{\min }|}{2} \ln \frac{\big (\sqrt{a^2+4|v_0^{\min }|}+a\big )^2 }{4|v_0^{\min }|}-d(a),\\ \Psi (r,t)&:=d(r)+\frac{r^2+L^2}{4}(2t+1)+\frac{|v_0^{\min }|}{2}\ln \left( 1+\frac{1}{t}\right) -Lr\sqrt{t(t+1)}. \end{aligned} \right\} \end{aligned}$$

(3.19)

The following proposition deals with a term similar to the hopping coefficient and plays a key role in the approximation of various error terms that we will encounter later, e.g. when we verify Assumption 2.1 for an electro-magnetic Schrödinger operator with multiple wells, as in Proposition 3.4.

Proposition 3.3

We have, as $h\rightarrow 0$,

$$\begin{aligned} w :=\int _{D(0,L)} {\mathfrak {u}}_h(x){\mathfrak {u}}_h(x_1+L,x_2)\mathrm e^{\frac{\mathrm iLx_2}{2h}}\mathop {}\!{\textrm{d}}x = {\mathcal {O}}\left( \mathrm e^{\frac{-S_a+o(1)}{h}}\right) , \end{aligned}$$

where $a=a(v_0)$, $L >2a$ and $S_a$ is defined in (3.9b).

Proof

We can estimate w in the same way as the hopping coefficient ${\mathfrak {c}}_h({\mathfrak {v}}_0,L)$ was estimated in [17, Prop. 3.5].

The only difference is that in the expression of ${\mathfrak {c}}_h(v_0,L)$ (see (3.15)) we encounter the potential energy term $v_0$ in the integrand and the integral is consequently over the smaller disc D(0, a). Hence the new term to estimate corresponds to $w_2$ below, $w_3$ being of the same type as $w_1$ after a symmetry.

We decompose the integral defining w into three terms

$$\begin{aligned} \begin{aligned} w&=w_1+w_2+w_3\,,\\ w_1&=\int _{D(0,a)} {\mathfrak {u}}_h(x){\mathfrak {u}}_h(x_1+L,x_2)\mathrm e^{\frac{\mathrm iLx_2}{2h}}\mathop {}\!{\textrm{d}}x\,, \\ w_2&=\int _{D(0,L-a)\setminus D(0,a)} {\mathfrak {u}}_h(x){\mathfrak {u}}_h(x_1+L,x_2)\mathrm e^{\frac{\mathrm iLx_2}{2h}}\mathop {}\!{\textrm{d}}x\,, \\ w_3&=\int _{D(0,L)\setminus D(0,L-a)} {\mathfrak {u}}_h(x){\mathfrak {u}}_h(x_1+L,x_2)\mathrm e^{\frac{\mathrm iLx_2}{2h}}\mathop {}\!{\textrm{d}}x\,. \end{aligned} \end{aligned}$$

Step 1: Contribution of the integral in D(0, a). We express the integral defining $w_1$ in polar coordinates

$$\begin{aligned} w_1 = \int _0^{2\pi }\int _0^{a} u_h(r) u_h\bigl (\sqrt{r^2+L^2+2Lr\cos \theta }\bigr ) \mathrm e^{\frac{\mathrm iLr\sin \theta }{2h}} r\mathop {}\!{\textrm{d}}r. \end{aligned}$$

(3.20)

In light of the decay property in (3.14), we get

$$\begin{aligned} w_1={\mathcal {O}}(h^{-1}\mathrm e^{-{\tilde{S}}/h}), \end{aligned}$$

where

$$\begin{aligned} {\tilde{S}} = \inf _{0<\theta<2\pi } \Big \{ \inf _{0<r<a} \bigl [ d(r)+d(\sqrt{r^2+L^2+2Lr\cos \theta }) \bigr ] \Big \}. \end{aligned}$$

Since $\sqrt{r^2+L^2+2Lr\cos \theta }>L-r$ and since $r\mapsto d(r)$ is non-decreasing,

$$\begin{aligned} {\tilde{S}} \ge \inf _{0<r<a}\bigl (d(r)+d(L-r) \bigr ) = d(a) + d(L-a) = S_a \end{aligned}$$

where $S_a$ is introduced in (3.9b). Notice that the penultimate identity follows from the fact that

$$\begin{aligned} \frac{\mathop {}\!{\textrm{d}}}{\mathop {}\!{\textrm{d}}r}[d(r) + d(L-r)] = \sqrt{\frac{r^2}{4}+v_0(r)-v_0^{\min }} - \sqrt{\frac{(L-r)^2}{4}-v_0^{\min }}< 0 \quad (0<r<a). \end{aligned}$$

Therefore, we have that

$$\begin{aligned} w_1 = {\mathcal {O}}(h^{-1}\mathrm e^{-S_a/h}). \end{aligned}$$

(3.21)

Step 2: Contribution of the integral in $D(0,L)\setminus D(0,L-a)$. We argue as in Step 1. Expressing the integral defining $w_3$ in polar coordinates

$$\begin{aligned} w_3 = \int _0^{2\pi }\int _{L-a}^{L} u_h(r) u_h\bigl (\sqrt{r^2+L^2+2Lr\cos \theta }\bigr ) \mathrm e^{\frac{\mathrm iLr\sin \theta }{2h}} r\mathop {}\!{\textrm{d}}r, \end{aligned}$$

(3.22)

we get

$$\begin{aligned} w_3={\mathcal {O}}(h^{-1}\mathrm e^{-{\tilde{S}}'/h}), \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} {\tilde{S}}'&=\inf _{0<\theta<2\pi }\left( \inf _{L-a<r<L}\Bigl (d(r)+d\bigl (\sqrt{r^2+L^2+2Lr\cos \theta }\bigr ) \Bigr )\right) \\&\ge \inf _{L-a<r<L}\bigl (d(r)+d(L-r) \bigr ) =\inf _{0<t<a}\bigl (d(t)+d(L-t) \bigr )=S_a. \end{aligned} \end{aligned}$$

Therefore, we have that

$$\begin{aligned} w_3 = {\mathcal {O}}(h^{-1}\mathrm e^{-S_a/h}). \end{aligned}$$

(3.23)

Step 3: Contribution of the integral in $D(0,L-a)\setminus D(0,a)$. We express the integral defining $w_2$ as follows

$$\begin{aligned} w_2 = \int _a^{L-a} u_h(r) \biggl (\int _0^{2\pi }K_h(r,\theta )d\theta \biggr )r\mathop {}\!{\textrm{d}}r\,, \end{aligned}$$

(3.24)

where

$$\begin{aligned} K_h(r,\theta )= u_h\left( \sqrt{r^2+L^2+2Lr\cos \theta }\right) e^{\frac{\mathrm iLr\sin \theta }{2h}}\,. \end{aligned}$$

Observing that for $a<r<L-a$, we have $a<L-r<L-a$ and

$$\begin{aligned} \rho :=\sqrt{r^2+L^2+2Lr\cos \theta }\ge L-r>a, \end{aligned}$$

so $u_h(\rho )$ has a nice integral representation [10, Eq. (2.9)]. Consequently, the integral of $K_h$ with respect to $\theta$ is computed as in [10, Prop. 5.1]. In fact, we have (see [17, Eq. (3.23)-(3.24)])

$$\begin{aligned} \int _0^{2\pi }K_h(r,\theta )d\theta = C_h\exp \left( -\frac{r^2+L^2}{4h}\right) \int _0^{+\infty } G_h(r,t) \mathop {}\!{\textrm{d}}t\,, \end{aligned}$$

(3.25)

where

$$\begin{aligned} G_h(r,t) = \exp \left( -\frac{(r^2+L^2)t}{2h} \right) t^{\alpha -1}(1+t)^{-\alpha }I_0\left( \frac{Lr\sqrt{t(t+1)}}{h}\right) , \end{aligned}$$

(3.26)

$C_h,\alpha$ are constants and $z\mapsto I_0(z)$ is the modified Bessel’s function of order 0 (see [17, Lem. 5.2]).

Let $\eta \in (0,1)$ and observe that [17, Lem. 4.1],

$$\begin{aligned}{} & {} \int _a^{L-a} u_h(r) \biggl (C_h\exp \biggl ( -\frac{r^2+L^2}{4h}\biggr )\int _0^{\eta } G_h(r,t) dt\biggr )r\mathop {}\!{\textrm{d}}r \\{} & {} \quad = {\mathcal {O}}(\mathrm e^{c\sqrt{\eta }/h})\int _a^{L-a} u_h(r)u_h(\sqrt{L^2+r^2}) r\mathop {}\!{\textrm{d}}r. \end{aligned}$$

By the decay properties of $u_h$ in (3.14), we get (using also (3.10) for the last estimate)

$$\begin{aligned} \int _a^{L-a} u_h(r) \biggl (C_h\exp \biggl ( -\frac{r^2+L^2}{4h}\biggr )\int _0^{\eta } G_h(r,t) dt\biggr )r\mathop {}\!{\textrm{d}}r&= {\mathcal {O}}(\mathrm e^{(c\sqrt{\eta }-S_0)/h}) \\&={\mathcal {O}}(\mathrm e^{-S_a/h}), \end{aligned}$$

for sufficiently small $\eta$.

Now we deal with the following integral

$$\begin{aligned} \int _a^{L-a} u_h(r) \biggl (C_h\exp \biggl ( -\frac{r^2+L^2}{4h}\biggr )\int _{\eta }^{+\infty } G_h(r,t) dt\biggr )r\mathop {}\!{\textrm{d}}r, \end{aligned}$$

which is asymptotically equivalent to [17, Lem. 4.4]

$$\begin{aligned} \frac{{\mathfrak {m}}( v_0)}{\sqrt{2\pi h}}\int _a^{L-a} \sqrt{r}\,a_0(r)\int _\eta ^{+\infty } g_0(t)\exp \biggl (-\frac{\Psi (r,t)-F(v_0)}{h}\biggr )\mathop {}\!{\textrm{d}}t\mathop {}\!{\textrm{d}}r, \end{aligned}$$

where $F(v_0),\Psi (r,t)$ are introduced in (3.19) and ${\mathfrak {m}}(v_0),g_0(t)$ are introduced in [17, Eq. (3.17) and (4.5)] as follows

$$\begin{aligned} \begin{aligned} \mathfrak {m}(v_0)&= \frac{\sqrt{1+2v_0''(0)}}{2\pi }\sqrt{\frac{2a|v_0^{\min }|}{\pi }} \frac{\left( a^2+4|v_0^{\min }| \right) ^{1/4} }{\sqrt{a^2+4|v_0^{\min }|}+a},\\ g_0(t)&= \frac{1}{t^{5/4}(t+1)^{1/4}} \left( 1+\frac{1}{t}\right) ^{\frac{1}{2}(\sqrt{1+2v_0''(0)}-1)}, \end{aligned} \end{aligned}$$

and the function $a_0$ is defined as follows

$$\begin{aligned} a_0(r)=\frac{1}{2}\sqrt{\frac{1+2v_0''(0)}{\pi }} \exp \left( -\int _0^{r}\frac{1}{2}\frac{d''(\rho )}{d'(\rho )}+\frac{1}{2\rho }-\frac{\sqrt{1+2v''_0(0)}}{2d'(\rho )}d\rho \right) . \end{aligned}$$

Of importance to us is that

$$\begin{aligned} -F({\mathfrak {v}}_0)+\inf _{(r,t)\in [a,L-a]\times {\mathbb {R}}_+}\Psi (r,t) \ge S_a, \end{aligned}$$

which follows by the same argument as used in the proof of [17, Prop. 4.5]; for convenience, we provide details in Appendix A.

We get eventually

$$\begin{aligned} w_2 = {\mathcal {O}}(\mathrm e^{(-S_a+o(1))/h}). \end{aligned}$$

(3.27)

With (3.21) and (3.23), this achieves the proof of the proposition. $\square$

3.3 Verifying Assumption 2.1—superposition of single well potentials

We study the specific case where the potential V is given by

$$\begin{aligned} V(x) = \sum _{k=1}^n {\mathfrak {v}}_0(x-z_k)\,, \end{aligned}$$

(3.28)

where ${\mathfrak {v}}_0$ is the non-positive radial function satisfying (3.5) and (we identify ${\mathbb {C}}$ and ${\mathbb {R}}^2$)

$$\begin{aligned} z_k :=\frac{L}{\sqrt{2-2\cos (2\pi /n)}}\mathrm e^{2\mathrm ik\pi /n},\quad L>0. \end{aligned}$$

(3.29)

The wells in this setting are the points $\{z_k\}_{1\le k\le n}$ which are selected so that

$$\begin{aligned} {{\,\textrm{dist}\,}}(z_k,z_{k+1}) = L. \end{aligned}$$

(3.30)

Let us verify that V satisfies (3.2c). Our construction of the points $z_k$ is such that $z_{k+1}=g z_k$, for $k\in {\mathbb {Z}}/n{\mathbb {Z}}$. Since ${\mathfrak {v}}_0$ is radial, we have

$$\begin{aligned} {\mathfrak {v}}_0(g^{-1}x-z_k) = v_0(|g^{-1}(x-g z_k)|) = {\mathfrak {v}}_0(x-z_{k+1}). \end{aligned}$$

Consequently, (3.2c) holds and $T_h$ commutes with M(g). We still have to check that Assumption 2.1 holds with the choice in Assumption 2.8 (3).

We introduce the functions

$$\begin{aligned} u_{h,k}(x) = \chi (x-z_k){\mathfrak {u}}_h(x-z_k)\mathrm e^{-\mathrm iz_k\cdot {\textbf{A}}(x)/h}\quad (1\le k\le n), \end{aligned}$$

(3.31)

where

• $(z_k)_{1\le k\le n}$ are the points introduced in (3.29);

• ${\textbf{A}}$ is the vector field introduced in (3.1);

• $\chi \in C_c^\infty ({\mathbb {R}}^2;[0,1])$ is a radial cut-off function satisfying $\chi =1$ on D(0, L) and ${{\,\textrm{supp}\,}}\chi \subset D(0,L+\eta )$ with $\eta \in (0,1)$ fixed arbitrarily.

The phase term in (3.31) is due to the effect of magnetic translation, which ensures that $\psi _{h,k}(x)={\mathfrak {u}}_h(x-z_k)\mathrm e^{-\mathrm iz_k\cdot {\textbf{A}}(x)/h}$ satisfies

$$\begin{aligned} {\mathcal {L}}_{h,k}^{\textrm{sw}} \psi _{h,k} = \lambda (h)\psi _{h,k} \quad \text {where}\quad {\mathcal {L}}_{h,k}^{\textrm{sw}} :=(-\mathrm ih\nabla -{\textbf{A}})^2+{\mathfrak {v}}_0(\cdot -z_k). \end{aligned}$$

(3.32)

The above constructions ensures that Assumption 2.8 (3) holds (since $z\cdot {\textbf{A}}(g_n^{-1}x)=(g_nz)\cdot {\textbf{A}}(x)$). Moreover, the next proposition shows that Assumption 2.1 holds too.

Proposition 3.4

Let $T_h$ and $\lambda (h)$ be as in (3.4) and (3.12) respectively. The conditions in Assumption 2.1 hold with the following choices:

• (a) $\{u_{h,1},\ldots ,u_{h,n}\}$ are as in (3.31);

• (b) any constants ${\mathfrak {S}}_1,{\mathfrak {S}}_2,{\mathfrak {S}}_3,p$ satisfying

  $$\begin{aligned} {\mathfrak {S}}_1\in (0,{\hat{S}}_a),\quad {\mathfrak {S}}_2\in (0, 2{\hat{S}}_a), \quad {\mathfrak {S}}_3\in (0, {\hat{S}}_a ),\quad p\in (0,1], \end{aligned}$$

  (3.33)

  where ${\hat{S}}_a$ is introduced in (3.9).

Proof

Step 1. We have by (3.31) and (3.32),

$$\begin{aligned} r_{h,k}:=T_{h}u_{h,k}= \Bigl (-h^2(\Delta \chi _k) + 2\mathrm ih(\nabla \chi _k)\cdot (-\mathrm ih\nabla -{\textbf{A}})+\chi _k\sum _{i\not =k}{\mathfrak {v}}_0(x-z_i) \Bigr ) \psi _{h,k}, \end{aligned}$$

(3.34)

where $\chi _k(x)=\chi (x-z_k)$.

Since $r_{h,k}$ is supported in $D(z_k,L+\eta )\setminus D(z_k,L-a)$, we get by using (3.14) and the decay of the ground state ${\mathfrak {u}}_h$ that

$$\begin{aligned} \Vert r_{h,k}\Vert = {\mathcal {O}}(h^{-1/2}\mathrm e^{-{\hat{S}}_a/h}). \end{aligned}$$

(3.35)

Step 2. We have

$$\begin{aligned} \Vert u_{h,k}\Vert ^2 = \Vert {\mathfrak {u}}_h\Vert ^2-\int _{{\mathbb {R}}^2}(1-\chi ^2)|{\mathfrak {u}}_{h}|^2\mathop {}\!{\textrm{d}}x. \end{aligned}$$

Since $1-\chi ^2$ is supported in ${\mathbb {R}}^2\setminus D(0,L)$, we get by the decay of ${\mathfrak {u}}_h$ that

$$\begin{aligned} \Vert u_{h,k}\Vert ^2 = 1 + {\mathcal {O}}(h^{-1}\mathrm e^{-2S_0/h} ) \end{aligned}$$

where $S_0$ is introduced in (3.9a).

Let us now consider $i\not =j$. We first inspect the case where $|z_i-z_j|=L$, which occurs only if $j=i\pm 1$. By a change of variable, we check that (thanks to the invariance by rotation)

$$\begin{aligned} \langle u_{h,i},u_{h,j}\rangle = {\left\{ \begin{array}{ll} \langle u_{h,1}, u_{h,2}\rangle &{}\text {if } j = i + 1\\ \overline{\langle u_{h,1}, u_{h,2}\rangle } &{}\text {if } j = i - 1. \end{array}\right. } \end{aligned}$$

By (3.31), the function $u_{h,1}\overline{u_{h,2}}$ is supported in $D(z_1,L+\eta )\cap D(z_2,L+\eta )$. Then, we perform the following decomposition

$$\begin{aligned} \langle u_{h,1}, u_{h,2}\rangle = {\mathcal {E}}_1+{\mathcal {E}}_2+{\mathcal {E}}_3 \end{aligned}$$

where,

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_1&:=\int _{D(z_1,L)}{\mathfrak {u}}_h(x-z_1){\mathfrak {u}}_h(x-z_2)\mathrm e^{-\mathrm i(z_1-z_2)\cdot {\textbf{A}}(x)}\mathop {}\!{\textrm{d}}x,\\ {\mathcal {E}}_2&:=- \int _{|x-z_1|<L}(1-\chi (x-z_2)){\mathfrak {u}}_h(x-z_1){\mathfrak {u}}_h(x-z_2)\mathrm e^{-\mathrm i(z_1-z_2)\cdot {\textbf{A}}(x)}\mathop {}\!{\textrm{d}}x\\&={\mathcal {O}}(h^{-1/2}\mathrm e^{- S_0/h})\\ {\mathcal {E}}_3&:=\int _{\begin{array}{c} L<|x-z_1|<L+\eta \\ 0<|x-z_2|<L+\eta \end{array}}u_{h,1}(x)\overline{u_{h,2}(x)}\mathop {}\!{\textrm{d}}x= {\mathcal {O}}(h^{-1/2}\mathrm e^{- S_0/h}). \end{aligned} \end{aligned}$$

In fact, in $D(z_1,L)\cap D(z_2,L+\eta )$, we have

$$\begin{aligned} u_{h,1}(x)\overline{u_{h,2}(x)}=\chi (x-z_2){\mathfrak {u}}_h(x-z_1){\mathfrak {u}}_h(x-z_2)\mathrm e^{-\mathrm i(z_1-z_2)\cdot {\textbf{A}}(x)}\,, \end{aligned}$$

with ${\mathfrak {u}}_h$ a radial function. By a change of variable (the translation $x\mapsto x-z_1$ followed by the rotation by the angle $\alpha$ defined by $z_1-z_2=|z_1-z_2|\mathrm e^{\mathrm i\alpha }$), we get

$$\begin{aligned} |{\mathcal {E}}_1| = \biggl |\int _{D(0,L)} {\mathfrak {u}}_h(x){\mathfrak {u}}_h(x_1+L,x_2)\mathrm e^{\frac{\mathrm iLx_2}{2h}}\mathop {}\!{\textrm{d}}x\biggr |, \end{aligned}$$

and by Proposition 3.3, we get

$$\begin{aligned} {\mathcal {E}}_1={\mathcal {O}}(h^{-1/2}\mathrm e^{ -( S_a+o(1)) /h} )+{\mathcal {O}}(h^{-1/2}\mathrm e^{-S_0 /h} )={\mathcal {O}}(\mathrm e^{-{\hat{S}}_a/h}), \end{aligned}$$

where in the last step we used the inequalities ${\hat{S}}_a<S_a<S_0$ from Proposition 3.1.

If $|z_i-z_j|\not =L$, then $\langle u_{h,i}\,,\,u_{h,j}\rangle ={\mathcal {O}}(h^{-1/2}\mathrm e^{ -( S_a+o(1)) /h} )+{\mathcal {O}}(h^{-1/2}\mathrm e^{-S_0 /h} )$ by a similar argument.

Step 3. Let us now estimate $\lambda _{n+1}(h)$ from below. Consider a partition of unity on ${\mathbb {R}}^2$, $\sum _{k=0}^{n}\zeta _k^2=1$, where

$$\begin{aligned} \text {for } 1\le k\le n,\quad \zeta _k=1 \text { on } D\left( z_k,\frac{L}{2}\right) ,\quad {{\,\textrm{supp}\,}}\zeta _k\subset D(z_k,L-a)\,, \end{aligned}$$

and

$$\begin{aligned} {{\,\textrm{supp}\,}}\zeta _0\subset \Omega _0:={\mathbb {R}}^2\setminus \bigcup _{k=1}^nD\left( z_k,\frac{L}{2}\right) . \end{aligned}$$

Then, we have,

$$\begin{aligned} {\mathfrak {v}}_0(x-z_i)\zeta _k(x)=0 \text { for } i\ne k\,,~1\le k\le n\,, \end{aligned}$$

and

$$\begin{aligned} {\mathfrak {v}}_0(x-z_i)\zeta _0(x)=0 \text { for } 1\le i\le n\,. \end{aligned}$$

Pick a function f in the domain of ${\mathcal {L}}_h$. We have the decomposition formula

$$\begin{aligned} \langle T_hf ,f \rangle{} & {} = \sum _{k=1}^n \langle {\mathcal {L}}_h^{\textrm{sw}} (\zeta _kf_h) ,\zeta _kf_h\rangle +\langle (-\mathrm ih\nabla -{\textbf{A}})^2(\zeta _0f_h),\zeta _0f_h\rangle \\{} & {} \quad -h^2\sum _{k=0}^{n}\Vert |\nabla \zeta _k|f_h \Vert ^2 -\lambda (h). \end{aligned}$$

By the min-max principle, we have

$$\begin{aligned} \forall j\ge 1, \quad \lambda _j(h)\ge \lambda _j\Bigl ((-\mathrm ih\nabla -{\textbf{A}})^2\bigoplus \oplus _{k=1}^n{\mathcal {L}}_{h}^{\textrm{sw}} \Bigr )-\lambda (h) -M_0 n h^2, \end{aligned}$$

where

$$\begin{aligned} M_0 = \max _{0\le k\le n}\Vert \nabla \zeta _k\Vert _\infty ^2. \end{aligned}$$

Finally, we use the well known asymptotics (see [17, Prop. A.1])

$$\begin{aligned} \lambda (h)=\lambda _1({\mathcal {L}}_h^{\textrm{sw}})= v_0^{\min }+E_1 h+{\mathcal {O}} (h^{3/2}),\quad \lambda _2({\mathcal {L}}_h^{\textrm{sw}})= v_0^{\min }+E_2 h+{\mathcal {O}} (h^{3/2}), \end{aligned}$$

where $E_1= \sqrt{1+2v_0''(0)}$ and $E_2>E_1$, and we conclude that

$$\begin{aligned} \lambda _{n+1}(h)\ge (E_2-E_1)h +{\mathcal {O}}(h^{3/2}). \end{aligned}$$

$\square$

The next proposition allows us to verify Assumption 2.11.

Proposition 3.5

With $J_0(h)=\langle T_hu_{h,1},u_{h,1}\rangle$ and $J_1(h)=\langle T_hu_{h,1},u_{h,2}\rangle$ we have

$$\begin{aligned} J_0(h)={\mathcal {O}}(h^{-1}\mathrm e^{-2{\hat{S}}_a(v_0,L)/h}) \end{aligned}$$

and

$$\begin{aligned} J_1(h) = \mathrm e^{-\mathrm i\phi _n/2h}{\mathfrak {c}}_h(v_0,L) +{\mathcal {O}}(h^{-1}\mathrm e^{-S_0(v_0,L)/h} ), \end{aligned}$$

where ${\mathfrak {c}}_h( v_0,L)$ is the hopping coefficient introduced in (3.15), ${\hat{S}}_a(v_0,L)$ and $S_0(v_0,L)$ are introduced in (3.9) and

$$\begin{aligned} \phi _n=-\frac{L^2\sin (2\pi /n)}{2-2\cos (2\pi /n)}. \end{aligned}$$

Moreover,

$$\begin{aligned} h\ln |J_1(h)|\underset{h\searrow 0}{\sim }- S( v_0,L), \end{aligned}$$

where $S(v_0,L)$ is introduced in (3.18).

Proof

From (3.34), we have

$$\begin{aligned} \begin{aligned} J_0(h) = \langle ({\mathcal {L}}_h-\lambda (h)) u_{h,1},u_{h,1}\rangle&= \int _{L-a\le |x-z_1|\le L+\eta } r_{h,1}(x)\overline{u_{h,1}(x) }\mathop {}\!{\textrm{d}}x \\&= {\mathcal {O}}(h^{-1}\mathrm e^{-2{\hat{S}}_a/h}). \end{aligned} \end{aligned}$$

We now move to estimate $J_1(h)$. First let us recall that the symmetry relations in Assumption 2.8 (3) imply that $u_{h,1}=M(g)u_{h,n}$ and⁵

$$\begin{aligned} J_1(h) = \langle T_hu_{h,1},u_{h,2}\rangle = \langle T_hu_{h,n},u_{h,1}\rangle = \overline{\langle T_hu_{h,1},u_{h,n}\rangle }. \end{aligned}$$

Similarly to the previous estimate of $J_0(h)$, by (3.32) and (3.34) we have

$$\begin{aligned} J_1(h) = J^{\textrm{app}}_{1}(h)+ {\mathcal {O}}(h^{-1}\mathrm e^{- S_0/h} ), \end{aligned}$$

where

$$\begin{aligned} J^{\textrm{app}}_{1}(h) = \int _{D(z_n,a) } {\mathfrak {v}}_0(x-z_n){\mathfrak {u}}_h(x-z_1){\mathfrak {u}}_h(x-z_n)\mathrm e^{-\mathrm i(z_n-z_1)\cdot {\textbf{A}}(x)/h}\mathop {}\!{\textrm{d}}x. \end{aligned}$$

Notice that $(z_n-z_1)\cdot {\textbf{A}}(x)=(z_n-z_1)\cdot {\textbf{A}}(x-z_n+z_1)$. By a translation, we get

$$\begin{aligned} J^{\textrm{app}}_{1}(h) = \mathrm e^{-\mathrm i(z_n-z_1)\cdot {\textbf{A}}(z_1)/h}\int _{D(0,a) } {\mathfrak {v}}_0(y){\mathfrak {u}}_h(y+z_n-z_1){\mathfrak {u}}_h(y)\mathrm e^{-\mathrm i(z_n-z_1)\cdot {\textbf{A}}(y)/h}\mathop {}\!{\textrm{d}}y. \end{aligned}$$

With $\ell _n=\frac{L}{\sqrt{2-2\cos (2\pi /n)}}$, we have $z_n=(\ell _n,0)$ and $z_1=\big (\ell _n\cos (2\pi /n),\ell _n\sin (2\pi /n)\big )$. Hence

$$\begin{aligned} (z_n-z_1)\cdot {\textbf{A}}(z_1)=z_n\cdot {\textbf{A}}(z_1)=\frac{\phi _n}{2}. \end{aligned}$$

Observing that $|z_n-z_1|=L$, we write $z_n-z_1=L\mathrm e^{\mathrm i\beta _n}$ where $\beta _n\in (0,\pi ]$. We denote by $R_{\beta _n}$ the rotation by $\beta _n$ around the origin. Noticing that

$$\begin{aligned} u\cdot {\textbf{A}}(R_{\beta _n}v)=(R_{\beta _n}^{-1}u)\cdot {\textbf{A}}(u)\quad (u,v\in {\mathbb {R}}^2), \end{aligned}$$

we get that

$$\begin{aligned} (z_n-z_1)\cdot {\textbf{A}}(R_{\beta _n} x) = \begin{pmatrix} L\\ 0 \end{pmatrix} \cdot \begin{pmatrix} -x_2/2\\ x_1/2 \end{pmatrix} = -\frac{Lx_2}{2}. \end{aligned}$$

The change of variable $y\mapsto x=R_{\beta _n}^{-1}y$ yields

$$\begin{aligned} J^{\textrm{app}}_{1}(h) = \mathrm e^{-\mathrm i\phi _n/2h}\int _{D(0,a) } {\mathfrak {v}}_0(x){\mathfrak {u}}_h(x_1+L,x_2){\mathfrak {u}}_h(x)\mathrm e^{\mathrm iLx_2/h}\mathop {}\!{\textrm{d}}y \end{aligned}$$

where we used that the functions ${\mathfrak {v}}_0,{\mathfrak {u}}_h$ are radial and that

$$\begin{aligned} |R_{\beta _n}x+z_n-z_1|=|x+R_{\beta _n}^{-1}(z_n-z_1)|=|(x_1+L,x_2)|. \end{aligned}$$

Now we have that $\mathrm e^{-\mathrm i\phi _n/2h}J^{\textrm{app}}_{1}(h)={\mathfrak {c}}_h(v_0,L)$ and by Proposition 3.2

$$\begin{aligned} h\ln |J_1^{\textrm{app}}(h)| \underset{h\searrow 0}{\sim }- S( v_0,L). \end{aligned}$$

The same asymptotics holds for $|J_1(h)|$ because $S_0(v_0,L)>S(v_0,L)$. $\square$

From now on, we work under the following assumption on $(v_0,L)$.

Assumption 3.6

With $a=a(v_0)$ introduced introduced in (3.6) and $L>2a$, the function $v_0$ satisfies $2{\hat{S}}_a(v_0,L)> S(v_0,L)$, where ${\hat{S}}_a(v_0,L)$ and $S(v_0,L)$ are introduced in (3.9) and (3.18) respectively.

Remark 3.7

By Proposition 3.1, Assumption 3.6 holds if $L\ge 4a(v_0)$.

We fix the choice of ${\mathfrak {S}}_1,{\mathfrak {S}}_2,{\mathfrak {S}}_3$ as in (3.33) but with the additional condition that

$$\begin{aligned} S(v_0,L)<2\min _{1\le j\le 3}{\mathfrak {S}}_j. \end{aligned}$$

This is possible under Assumption 3.6 since $S(v_0,L) < 2 {\hat{S}}_a(v_0,L)$. Therefore, (2.22) holds with $S=S(v_0,L)$ and Proposition 3.5 ensures that the other conditions in Assumption 2.11 hold too⁶.

3.4 The case $n=2$: Double wells

We choose here V to be the double well potential defined by (3.28) for $n=2$. We therefore have two wells

$$\begin{aligned} z_1 = \Big (-\frac{L}{2},0\Big ),\quad z_2=\Big (\frac{L}{2},0\Big ),\quad L>2a>0. \end{aligned}$$

(3.36)

A straightforward application of (2.26), Propositions 3.4 and 3.5 (and Section 2.4.2) yields the following.

Theorem 3.8

Assuming the conditions in (3.5) and in Assumption 3.6 are fulfilled, then the following asymptotics holds,

$$\begin{aligned} h \ln \big (\lambda _2(h)-\lambda _1(h)\big ) \underset{h\searrow 0}{\sim } - S(v_0,L), \end{aligned}$$

(3.37)

where $S(v_0,L)$ is the constant introduced in (3.18).

3.4.1 Discussion

Let us introduce the following classes⁷ of admissible $(v_0,L)$, where $v_0$ satisfies the conditions in (3.5), and

$$\begin{aligned} \begin{aligned} {\mathscr {A}}&=\{(v_0,L)~:~L>2a(v_0) \text{ and } \text{( }3.37\text{) } \text{ holds }\},\\ {\mathscr {A}}^{\textrm{FSW}}&=\{(v_0,L)~:~L>4(a(v_0)+\sqrt{|v_0^{\min }|})\},\\ \overline{{\mathscr {A}}}^{\textrm{HKS}}&=\{(v_0,L)~:~v_0 \text{ satisfies } \text{ Assumption } 3.6\}\\ \underline{{\mathscr {A}}}^{\textrm{HKS}}&=\{(v_0,L)~:~L\ge 4a(v_0)\}. \end{aligned} \end{aligned}$$

By Proposition 3.1, we have

$$\begin{aligned} \underline{{\mathscr {A}}}^{\textrm{HKS}}\subset \overline{{\mathscr {A}}}^{\textrm{HKS}} \end{aligned}$$

and by Theorem 3.8

$$\begin{aligned} \overline{{\mathscr {A}}}^{\textrm{HKS}}\subset {\mathscr {A}}. \end{aligned}$$

The earlier results in [10, 17] yield that

$$\begin{aligned} {\mathscr {A}}^{\textrm{FSW}}\subset {\mathscr {A}}. \end{aligned}$$

However, our results are much stronger since ${\mathscr {A}}^{\textrm{FSW}}$ is a proper subset of $\underline{{\mathscr {A}}}^{\textrm{HKS}}$.

3.4.2 Proof of Theorem 1.2

In light of (3.3), it suffices to apply Theorem 3.8 with the effective semi-classical parameter $\hbar =b^{-1}h$ and the effective potential defined by $b^{-2}v_0$. We then obtain ${\mathscr {E}}_{b,L}(v_0)=bS(b^{-2}v_0,L)$.

3.5 The case $n=3$: Three wells and flux effects

Let us assume that $n=3$ so that V defined by (3.28) is a potential with three wells

$$\begin{aligned} z_1 = \frac{L}{2\sqrt{3}}(-1,\sqrt{3}), \quad z_2=\frac{L}{2\sqrt{3}}(-1,-\sqrt{3}),\quad z_3= \frac{L}{\sqrt{3}}(1,0), \end{aligned}$$

which are located on the vertices of an equilateral triangle with side length L and let $D\subset {\mathbb {R}}^2$ be its interior. The area of D is then $\frac{\sqrt{3}}{4}L^2$. The flux of the magnetic field in D is

$$\begin{aligned} \Phi :=\frac{1}{2\pi }\int _D{{\,\textrm{curl}\,}}{\textbf{A}}\mathop {}\!{\textrm{d}}x = \frac{\sqrt{3}L^2}{8\pi }. \end{aligned}$$

(3.38)

By Proposition 3.5, we have and

$$\begin{aligned} J_0(h) = {\mathcal {O}}(\mathrm e^{-2{\hat{S}}_a/h}) \text{ with } S({\mathfrak {v}}_0,L)<2{\hat{S}}_a\,. \end{aligned}$$

and

$$\begin{aligned} |J_1(h)|\underset{h\searrow 0}{\sim }|{\mathfrak {c}}_h(v_0,L)|=\exp \left( -\frac{ S({\mathfrak {v}}_0,L)+o(1)}{h} \right) . \end{aligned}$$

Recall that it was proved in [10] that ${\mathfrak {c}}_h(v_0,L)$ is a negative real number. So we infer from Proposition 3.5,

$$\begin{aligned} \mathrm e^{-2\pi \mathrm i\Phi /h}J_1(h)\underset{h\searrow 0}{\sim }{\mathfrak {c}}_h(v_0,L) \quad \text{ and } \quad {\mathfrak {c}}_h(v_0,L) \underset{h\searrow 0}{\sim }\ - |J_1(h)|. \end{aligned}$$

Therefore, we conclude that

$$\begin{aligned} J_1(h) \underset{h\searrow 0}{\sim }\ - |J_1(h)|\mathrm e^{2\pi \mathrm i\Phi /h}{} & {} =|J_1(h)|\mathrm e^{2\pi \mathrm i\Phi /h+\mathrm i\pi }, \quad |J_1(h)|\\{} & {} \underset{h\searrow 0}{=}\ \exp \left( -\frac{ S({\mathfrak {v}}_0,L)+o(1)}{h} \right) . \end{aligned}$$

Recall that by (2.27),

$$\begin{aligned} I_1(h) \underset{h\searrow 0}{\sim }\ J_1(h), \end{aligned}$$

(3.39)

and that $I_1(h)=|I_1(h)|\mathrm e^{\mathrm i\theta (h)}$, with $\theta (h)\in [0,2\pi )$. It results from (3.39) that $|I_1(h)|\sim |J_1(h)|$, and also

$$\begin{aligned} \mathrm e^{\mathrm i\theta (h)-2\pi \mathrm i\Phi /h-\mathrm i\pi }\sim 1. \end{aligned}$$

(3.40)

If for a given constant $c_0\in (0,1)$ we introduce the set

$$\begin{aligned} {\mathcal {N}}(c_0) = \left\{ h\in (0,1],~{{\,\textrm{dist}\,}}\left( \frac{\Phi }{h}+\frac{1}{2},{\mathbb {Z}}\right) \ge c_0\right\} , \end{aligned}$$

(3.41)

then

$$\begin{aligned} \forall h\in {\mathcal {N}}(c_0),\quad 0<c_0\le \frac{\Phi }{h}+\frac{1}{2}-\left\lfloor \frac{\Phi }{h}+\frac{1}{2}\right\rfloor \le 1-c_0<1, \end{aligned}$$

and it results from (3.40) that (since $\theta (h)\in [0,2\pi )$ in the definition of $I_1(h)$)

$$\begin{aligned} \theta (h)\underset{\begin{array}{c} h\searrow 0\\ h\in {\mathcal {N}}(c_0) \end{array}}{=} 2\pi \left( \frac{\Phi }{h}+\frac{1}{2}-\left\lfloor \frac{\Phi }{h}+\frac{1}{2}\right\rfloor \right) +o(1). \end{aligned}$$

(3.42)

Now by paragraph 2.4.3 we get crossing of eigenvalues quantified via the following functions

$$\begin{aligned} \begin{aligned} {\textsf{a}}(x)&=\cos \left( x+\frac{2\pi }{3}\right) -\cos \left( x\right) ,\\ {\textsf{b}}(x)&=\cos \left( x+\frac{4\pi }{3}\right) -\cos \left( x+\frac{2\pi }{3}\right) . \end{aligned} \end{aligned}$$

(3.43)

Theorem 3.9

Assume that the conditions in (3.5) are fulfilled and that V is defined by (3.28) with $n=3$. If Assumption 3.6 holds, then there is a relabeling $\mu _1(h),\mu _2(h),\mu _3(h)$ of the eigenvalues $\lambda _1(h),\lambda _2(h),\lambda _3(h)$ of the electro-magnetic Schrödinger operator ${\mathcal {L}}_h$ such that the asymptotics

$$\begin{aligned} \begin{aligned} \mu _2(h)-\mu _1(h)&\underset{h\searrow 0}{=}\ \bigl ({\textsf{a}}(\Phi /h)+o(1)\bigr ) \exp \left( \frac{- S( v_0,L)+o(1)}{h}\right) \\ \mu _3(h)-\mu _2(h)&\underset{h\searrow 0}{=}\bigl ({\textsf{b}}(\Phi /h) +o(1)\bigr ) \exp \left( \frac{- S( v_0,L)+o(1)}{h}\right) \end{aligned} \end{aligned}$$

(3.44)

hold for all $L>2a(v_0)$.

Moreover, there exists a sequence $\bigl (h_1(k),h_2(k),h_3(k)\bigr )_{k\ge k_0}$ which converges to 0 such that, for all $k\ge k_0$ we have

$$\begin{aligned} 0<h_1(k+1)<h_3(k)<h_2(k)<h_1(k)<1 \end{aligned}$$

and

$$\begin{aligned} \mu _1\big (h_1(k)\big )=\mu _3\big (h_1(k)\big ),~ \mu _1\big (h_2(k)\big )=\mu _2\big (h_2(k)\big ),~ \mu _2\big (h_3(k)\big )=\mu _3\big (h_3(k)\big ). \end{aligned}$$

Proof

To get the asymptotics in (3.44), we use the computations in Paragraph 2.4.3 and (3.39). Then we approximate $|J_1(h)|$ by Proposition 3.5.

As indicated in (3.42), the function $h\mapsto \theta (h)$ will not be continuous on any open right neighborhood of 0, since, as h decreases towards 0, it will necessarily make infinite many jumps of size $2\pi$. Given $\epsilon > 0$ we can, however, avoid the jumps by restricting $\theta$ to the union of an infinite number of small disjoint intervals $\mathcal {I}_k$, $k \in \mathbb {N}$, with the right endpoint of $\mathcal {I}_{k + 1}$ is less than the left endpoint of $\mathcal {I}_k$, and moreover such that $\theta$ maps each such interval continuously onto the interval $(\epsilon ,2\pi - \epsilon )$. This follows from (3.40) and the fact that $h \mapsto I_1(h)$ is continuous. In each interval $\mathcal {I}_k$ we can find $h_1(k)$, $h_2(k)$ and $h_3(k)$ such that

$$\begin{aligned} \theta (h_1(k)) = \frac{\pi }{3},\quad \theta (h_2(k)) = \frac{2\pi }{3},\quad \theta (h_3(k)) = \pi . \end{aligned}$$

In fact, since $h \mapsto \Phi /h + 1/2$ is strictly monotone and since $\pi /3$, $2\pi /3$ and $\pi$ are well separated, we can, eventually by starting our counting on a larger k (which means that we consider smaller h), impose that $h_1(k)> h_2(k) > h_3(k)$ holds. We conclude that we get the eigenvalue crossings as indicated in Paragraph 2.4.3. $\square$

4 Smoothed triangles and Neumann boundary condition

4.1 Introduction

4.1.1 Geometric setting

In this section, $\Omega$ is a bounded open set of ${\mathbb {R}}^2$ with $C^\infty$ boundary $\Gamma$. The Hilbert space is $H=L^2(\Omega )$ and we will assume (see below for the invariance assumption) that we are in the situation considered in Remark 2.7.

We assume that $\Gamma$ is a simple curve and denote its length by $|\Gamma |=2L$. Let ${\mathbb {R}}/2L{\mathbb {Z}}\ni s\mapsto \gamma (s)$ be the arc-length parameterization of $\Gamma$ such that the unit tangent vector $\tau (s):={\dot{\gamma }}(s)$ turns counterclockwise. Let us denote by k the curvature along $\Gamma$ defined as follows

$$\begin{aligned} \ddot{\gamma }(s)=k(s)\nu (s) \end{aligned}$$

where $\nu (s)$ is the unit normal to $\Gamma$ at $\gamma (s)$ pointing inward $\Omega$.

Definition 4.1

We introduce the maximal curvature along $\Gamma$ as follows

$$\begin{aligned} k_{\max }=\max _{s\in {\mathbb {R}}/2L{\mathbb {Z}}} k(s) \end{aligned}$$

(4.1)

and call a point $z_0\in \Gamma$ a curvature well if $z_0=\gamma (s_0)$ and $k(s_0)=k_{\max }$. The point $z_0$ is said to be a non-degenerate curvature well if furthermore $k''(s_0)<0$. We denote by $\Gamma _0$ the set of curvature wells.

Consider a positive integer n and the rotation by $2\pi /n$ denoted by $g_n$. We assume that

$$\begin{aligned} \Omega \text { is invariant by the rotation } g_n \end{aligned}$$

(4.2a)

and that we have n non-degenerate curvature wells

$$\begin{aligned} \Gamma _0=\{z_1=\gamma (s_1),\cdots ,z_n=\gamma (s_n)\} \end{aligned}$$

(4.2b)

with⁸

$$\begin{aligned} s_j=(j-1)\frac{2L}{n}, \quad k''(s_j)<0\quad (1\le j\le n). \end{aligned}$$

(4.2c)

The symmetry assumption yields that

$$\begin{aligned} z_{j+1}=g_nz_j\quad (1\le j\le n-1). \end{aligned}$$

(4.2d)

and

$$\begin{aligned} \gamma \left( s+\frac{2L}{n}\right) = g_n\gamma (s),\quad k\left( s+\frac{2L}{n}\right) =k(s)\quad (s\in {\mathbb {R}}/2\pi {\mathbb {Z}}). \end{aligned}$$

(4.2e)

4.1.2 The magnetic Neumann Laplacian

We are interested in the magnetic Neumann Laplacian ${\mathscr {L}}_h^N=(-\mathrm ih\nabla -{\textbf{A}})^2$, in the Hilbert space $L^2(\Omega )$, where the magnetic field $B={{\,\textrm{curl}\,}}{\textbf{A}}=1$ is uniform⁹ and generated by the magnetic potential ${\textbf{A}}$ introduced in (3.1). The operator ${\mathscr {L}}_h^N$ acts on functions $u\in H^2(\Omega )$ satisfying the (magnetic) Neumann condition

$$\begin{aligned} \nu \cdot (-\mathrm ih\nabla -{\textbf{A}})u|_{\Gamma }=0. \end{aligned}$$

The case of double curvature wells corresponding to $n=2$ was treated in [6], where the symmetry was generated by the reflection ${\tilde{g}}_2$ (see Remark 2.9). Here we focus on the case with $n\ge 3$ curvature wells and where the symmetry is generated by a rotation. When $n=3$, a typical example is the smoothed triangle (Fig. 2). When $\Omega$ is an equilateral triangle the heuristic discussion is given in [11] but no rigorous result can be given since the authors were unable to have a sufficiently accurate control of the tunneling. Numerically, this has been computed in [4] which in particular gives the enlightening picture predicting eigenvalue crossings (Fig. 4). We refer to [5, Sec. 5] for rigorous analysis in the case of polygons and to [27] for a comprehensive study of the magnetic Laplacian.

Fig. 4

Eigenvalue crossings in an equilateral triangle

The investigation of ${\mathscr {L}}_h^N$ can be connected with Section 2 after shifting by a constant $\ell (h)$ and taking the operator $T_h$ as follows

$$\begin{aligned} T_h = {\mathscr {L}}_h^N-\ell (h). \end{aligned}$$

(4.3)

The shift constant $\ell (h)$ will be defined by the ground state energy of an operator with a single curvature well.

Two constants are important in our analysis. First we meet a magnetic flux like term that controls the eigenvalue crossings and is defined by

$$\begin{aligned} \Phi _0 = |\Omega |-|\Gamma |\xi _0=|\Omega |-2L\xi _0, \end{aligned}$$

(4.4a)

Secondly the following constant controls the strength of the tunneling and is defined by

$$\begin{aligned} {\textsf{S}}_n=\sqrt{ \frac{2C_1}{(\mu _1^N)''(\xi _0)}}\int _0^{2L/n} \sqrt{k_{\max }-k(s)}\mathop {}\!{\textrm{d}}s. \end{aligned}$$

(4.4b)

The definition of $\Phi _0$ and ${\textsf{S}}_n$ involves universal constants related to the de Gennes model. We recall that, for $\xi \in {\mathbb {R}}$, $\mu _1^N(\xi )$ denotes the lowest eigenvalue of the Neumann realization of the Harmonic oscillator on the semi-axis ${\mathbb {R}}_+$:

$$\begin{aligned} {\mathfrak {h}}^{N}[\xi ]=-\frac{d^2}{d\tau ^2}+(\xi +\tau )^2 \end{aligned}$$

(4.5a)

Minimizing over $\xi \in {\mathbb {R}}$ we get the de Gennes constant

$$\begin{aligned} \Theta _0 = \inf _{\xi \in {\mathbb {R}}}\mu ^{N}_1(\xi )=\mu ^{N}_1(\xi _0), \text{ where } \xi _0=-\sqrt{\Theta _0}\,. \end{aligned}$$

(4.5b)

Denoting by $u_0$ the positive $L^2$- normalized ground state of ${\mathfrak {h}}^N[\xi _0]$, the constant $C_1$ appearing in (4.4b) is defined by

$$\begin{aligned} C_1 = \frac{|u_0(0)|^2}{3}\,. \end{aligned}$$

(4.5c)

4.2 Reduction to tubular domain

In a tubular neighborhood of the boundary, we can work with adapted coordinates $(s,t)\in {\mathbb {R}}/2L{\mathbb {Z}}\times {\mathbb {R}}_+$ that are linked to Cartesian coordinates as follows

$$\begin{aligned} (x_1,x_2)={\mathscr {T}}(s,t):=\gamma (s)-t\nu (s). \end{aligned}$$

(4.6)

There exists a geometric constant $\varepsilon _0>0$ such that the above transformation is invertible when $0<t<\varepsilon _0$; the image of $\Phi$ is the tubular neighborhood of $\Gamma$

$$\begin{aligned} \Gamma (\varepsilon _0) = \{x\in \Omega ~:~\textrm{dist}(x,\partial \Omega )<\varepsilon _0\} \end{aligned}$$

(4.7)

and for $x\in \Gamma (\varepsilon _0)$, $(s,t)={\mathscr {T}}^{-1}(x)$ means that s is the arc-length coordinate of the projection of x on $\Gamma$ and t is the normal distance from x to $\Gamma$. We will refer to s as the tangential variable and to t as the normal variable.

Let us understand the action of the operator in (2.14) in these adapted coordinates. Let u be supported in $\Gamma (\varepsilon _0)$ and $v=M(g)u$. Then by (4.2e), ${\tilde{v}}(s,t)=v\circ {\mathscr {T}}$ satisfies

$$\begin{aligned} {\tilde{v}}(s,t) = {\tilde{u}}\left( s-\frac{2L}{n},t \right) ,\quad {\tilde{u}}=u\circ {\mathscr {T}}. \end{aligned}$$

(4.8)

The Hilbert space $L^2(\Gamma (\varepsilon _0))$ is transformed to the weighted space

$$\begin{aligned} L^2\big (({\mathbb {R}}/ 2L {\mathbb {Z}}) \times (0,\varepsilon _0);a\mathop {}\!{\textrm{d}}s \mathop {}\!{\textrm{d}}t\big ),\quad a(s,t)=1-tk(s). \end{aligned}$$

After a gauge transformation to eliminate the normal component of ${\textbf{A}}$, see [11, App. F], the action of the operator ${\mathscr {L}}_h^N$ is transformed into

$$\begin{aligned} \begin{aligned} \tilde{{\mathscr {L}}}_h&:=-h^2 a^{-1}\partial _t a\partial _t + a^{-1}\left( -\mathrm ih\partial _s+\gamma _0-t+\frac{k(s)}{2}t^2\right) \\&\qquad \times a^{-1}\left( -\mathrm ih\partial _s+\gamma _0-t+\frac{k(s)}{2}t^2\right) \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \gamma _0=\frac{|\Omega |}{2L}=\frac{\pi }{L} \Phi \end{aligned}$$

(4.9)

and

$$\begin{aligned} \Phi =\frac{1}{2\pi }\int _\Omega B\mathop {}\!{\textrm{d}}x \text{ is } \text{ the } \text{ magnetic } \text{ flux } \text{ in } \Omega \,. \end{aligned}$$

The change of variables, $t=h^{1/2}\tau$ and $s=\sigma$, transforms ${\mathbb {R}}/ 2L {\mathbb {Z}}\times (0,\varepsilon _0)$ and the measure to $a\mathop {}\!{\textrm{d}}s\mathop {}\!{\textrm{d}}t$ to

$$\begin{aligned} {\tilde{\Gamma }}_h=({\mathbb {R}}/ 2L {\mathbb {Z}})\,\times (0,\varepsilon _0 h^{-1/2})\quad \textrm{and }\quad {\tilde{a}}_h(\sigma ,\tau )=1-h^{1/2}\tau k(\sigma ). \end{aligned}$$

Moreover, it transforms the Hilbert space $L^2\big (({\mathbb {R}}/ 2L {\mathbb {Z}})\,\times (0,\varepsilon _0);a\mathop {}\!{\textrm{d}}s\mathop {}\!{\textrm{d}}t\big )$ to the Hilbert space $L^2\big ({\tilde{\Gamma }}_h;{\tilde{a}}_h\mathop {}\!{\textrm{d}}\sigma \mathop {}\!{\textrm{d}}\tau \big ),$ and the operator $\tilde{{\mathscr {L}}}_h$ to $h\tilde{{\mathscr {N}}}_h$ where

$$\begin{aligned} \tilde{{\mathscr {N}}}_{h}= & {} -{\tilde{a}}^{-1}_{h}\partial _\tau {\tilde{a}}_{h}\partial _\tau +\tilde{a}^{-1}_{h}\left( -\mathrm ih^{1/2}\partial _\sigma +{h^{-1/2}\gamma _0}-\tau +h^{1/2} \frac{k(\sigma )}{2}\tau ^2\right) \nonumber \\{} & {} \times \tilde{a}^{-1}_{h}\left( -\mathrm ih^{1/2}\partial _\sigma +{ h^{-1/2}\gamma _0}-\tau +h^{1/2} \frac{k(\sigma )}{2}\tau ^2\right) . \end{aligned}$$

(4.10)

Let us consider $\tilde{{\mathscr {N}}}_h$, in $L^2\big ({\tilde{\Gamma }}_h;{\tilde{a}}_h\mathop {}\!{\textrm{d}}\sigma d\tau \big )$, with domain¹⁰

$$\begin{aligned} \begin{aligned} \textsf{Dom}(\tilde{{\mathscr {N}}}_h) = \{u\in L^2\big ({\tilde{\Gamma }}_h;d\sigma d\tau \big )~:~&\partial _\tau ^2u,\partial _\sigma ^2u\in L^2\big ({\tilde{\Gamma }}_h;d\sigma d\tau \big ),\\&\partial _\tau u|_{\tau =0}=0,~u|_{\tau =\varepsilon _0h^{-1/2}}=0\}. \end{aligned} \end{aligned}$$

The study of the eigenvalues of ${\mathscr {L}}_h^N$, can then be compared with those of $\tilde{{\mathscr {N}}}_h$ (see [6, Prop. 2.7]).

Proposition 4.2

Let $N\in {\mathbb {N}}$ and ${\textsf{S}}_n$ be the constant introduced in (4.4b). There exist $K>\textsf{S}_n$, $C,h_0>0$ such that, for all $h\in (0,h_0]$ and $1\le k\le N$, we have,

$$\begin{aligned} \lambda _k ({\mathscr {L}}_h^N)-Ce^{-K/h^{\frac{1}{4}}}\,\le h\lambda _k(\tilde{{\mathscr {N}}}_h)\le \lambda _k({\mathscr {L}}_h^N)+Ce^{-K/h^{\frac{1}{4}}}\,, \end{aligned}$$

where $\lambda _k({\mathscr {L}}_h^N)$ and $\lambda _k(\tilde{{\mathcal {N}}}_h)$ are the k-th eigenvalues, counting multiplicity, of the operators ${\mathscr {L}}_h^N$ and $\tilde{{\mathscr {N}}}_{h}$, respectively.

Recall that we are interested in applying the results in Sect. 2 to the operator $T_h$ obtained by shifting the operator ${\mathscr {L}}_h^N$ (see (4.3)). Effectively, that is related to the operator obtained by doing the corresponding shift to the operator $\tilde{{\mathscr {N}}}_h$,

$$\begin{aligned} {\tilde{T}}_h = \tilde{{\mathscr {N}}}_h-h^{-1}\ell (h). \end{aligned}$$

(4.11)

Our next task is to verify Assumptions 2.1 and 2.8 (they will both hold for ${\tilde{T}}_h$ and $T_h$) and this will require the construction of certain quasi-modes $(u_{h,i})_{1\le i\le n}$. Let us make the following two observations:

• (i) The assumption in (4.2a) implies that ${\tilde{T}}_h$ commutes with $M(g_n)$, hence the condition of invariance by rotation holds.

• (ii) For every fixed labeling n, the eigenfunctions of ${\tilde{T}}_h$ corresponding to the n’th eigenvalue decay exponentially in the (rescaled) tangential variable; more precisely, given an eigenfunction $f_n$ of ${\tilde{T}}_h$ with corresponding eigenvalue $\lambda _n({\tilde{T}}_h$, there exist positive constants $C_n,h_n,\alpha _n$ such that

  $$\begin{aligned} \forall \,h\in (0,h_n],\quad \int _{{\tilde{\Gamma }}_h}\mathrm e^{\alpha _k\tau }\bigl (|\partial _\tau f_n|^2+|f_n|^2 \bigr )\mathop {}\!{\textrm{d}}s\mathop {}\!{\textrm{d}}\tau \le C_k\int _{{\tilde{\Gamma }}_h}|f_n|^2\mathop {}\!{\textrm{d}}s\mathop {}\!{\textrm{d}}\tau . \end{aligned}$$

The relevance of (i) above is that we can construct quasi-modes of ${\tilde{T}}_h$ obeying the symmetry invariance properties as in Assumption 2.8, and in turn we can use these quasi-modes to produce quasi-modes for the initial operator $T_h$ in (4.3). The observation in (ii) asserts that the eigenfunctions of ${\tilde{T}}_h$, once rescaled to the initial tangential variable $t=h^{1/2}\tau$, can be ignored in the interior of the domain $\Omega$.

The expression of the operator $\tilde{{\mathscr {N}}}_h$ (and hence ${\tilde{T}}_h$) involves the effective semi-classical parameter $\hbar :=h^{1/2}$. This leads us to adjust our setting by working with the operators

$$\begin{aligned} {\mathscr {T}}_\hbar = {\mathscr {N}}_\hbar -\lambda (\hbar ) \end{aligned}$$

(4.12)

and

$$\begin{aligned} {\mathscr {N}}_{\hbar }= & {} -a^{-1}_{\hbar }\partial _\tau a_{\hbar }\partial _\tau +{a}^{-1}_{\hbar }\left( -\mathrm i\hbar \partial _\sigma +{ \hbar ^{-1}\gamma _0}-\tau +\hbar \frac{k(\sigma )}{2}\tau ^2\right) \nonumber \\{} & {} \times {a}^{-1}_{\hbar }\left( -\mathrm i\hbar \partial _\sigma +{ \hbar ^{-1}\gamma _0}-\tau +\hbar \frac{k(\sigma )}{2}\tau ^2\right) \end{aligned}$$

(4.13)

where

$$\begin{aligned} a_\hbar (\sigma ,\tau ) = 1-\hbar \tau k(\sigma ) = {\tilde{a}}_h(\sigma ,\tau ), \quad \lambda (\hbar ) = h^{-1}\ell (h)\,. \end{aligned}$$

4.3 The single well problem

4.3.1 Definition of the operator

Let us recall that by (4.2a) and (4.2b), we have on the interval $(-2L/n,2L/n)$ a single non-degenerate maximum, $s_1=0$, of the curvature k(s). Let us fix a positive $\eta <\min (\frac{1}{4},\frac{L}{4n})$ and consider the following set

$$\begin{aligned} {\tilde{\Gamma }}_{\hbar ,\eta }^{(1)}=\Bigl (-\frac{2L}{n}+\eta ,\frac{2L}{n}-\eta \Bigr )\times (0,\varepsilon _0\hbar ^{-1}). \end{aligned}$$

Now we consider the operator

$$\begin{aligned} {\mathscr {N}}_{\hbar }^{(1)}= & {} -a^{-1}_{\hbar }\partial _\tau a_{\hbar }\partial _\tau +{a}^{-1}_{\hbar }\left( -\mathrm i\hbar \partial _\sigma +{ \hbar ^{-1}\gamma _0}-\tau +\hbar \frac{k(\sigma )}{2}\tau ^2\right) \nonumber \\{} & {} \times {a}^{-1}_{\hbar }\left( -\mathrm i\hbar \partial _\sigma +{ \hbar ^{-1}\gamma _0}-\tau +\hbar \frac{k(\sigma )}{2}\tau ^2\right) \end{aligned}$$

(4.14)

with domain

$$\begin{aligned} \begin{aligned} \textsf{Dom}({\mathscr {N}}_{\hbar }^{(1)})=\{u\in L^2\big ({\tilde{\Gamma }}_{\hbar ,\eta }^{(1)};d\sigma d\tau \big )~:~&\partial _\tau ^2u,\partial _\sigma ^2u\in L^2\big ({\tilde{\Gamma }}_{\hbar ,\eta }^{(1)};d\sigma d\tau \big ),\\&\partial _\tau u|_{\tau =0}=0,~u|_{\tau =\varepsilon _0h^{-1/2}}\\&=0,~ u|_{\sigma =\pm ( 2L/n-\eta )}=0\}. \end{aligned} \end{aligned}$$

We denote by $\lambda (\hbar )$ the ground state energy of the operator ${\mathscr {N}}_{\hbar }^{(1)}$; it is simple and can be expanded as follows [7, 12]

$$\begin{aligned} \lambda (\hbar )\underset{\hbar \rightarrow 0}{\sim }\Theta _0-3C_1k_{\max }\hbar +C_1\Theta _0^{1/4}\sqrt{\frac{3|k_2|}{2}}\hbar ^{3/2}+\sum _{j\ge 4}\delta _{1,j}\hbar ^{j/2}. \end{aligned}$$

(4.15a)

where $(\delta _{1,j})_{j\ge 1}$ are real constants and $k_2=k_2''(0)<0$. Moreover there exist real constants $(\delta _{2,j})_{j\ge 1}$ such that the second eigenvalue satisfies

$$\begin{aligned} \lambda _2(\hbar )\underset{\hbar \rightarrow 0}{\sim }\Theta _0-3C_1k_{\max }\hbar +3C_1\Theta _0^{1/4}\sqrt{\frac{3|k_2|}{2}}\hbar ^{3/2}+\sum _{j\ge 4}\delta _{2,j}\hbar ^{j/2} \end{aligned}$$

(4.15b)

and there is a spectral gap

$$\begin{aligned} \lambda _2(\hbar ) -\lambda (\hbar ) \underset{\hbar \rightarrow 0}{\sim }\ 2 C_1\Theta _0^{1/4}\sqrt{\frac{3|k_2|}{2}}\hbar ^{3/2}. \end{aligned}$$

Functions in the domain of ${\mathscr {N}}_{\hbar }^{(1)}$ can be extended to the full half-plane ${\mathbb {R}}_+^2={\mathbb {R}}\times {\mathbb {R}}_+$ after multiplication by a suitable cutoff function. We could have considered the single well problem in an alternative manner by truncating the curvature and extending it by 0 outside ${\hat{\Gamma }}_{\hbar ,\eta }^{(1)}$ and also the weight function $a_\hbar$ to get an operator in ${\mathbb {R}}^2_+$. Due to the exponential decay of bound states, the spectra agree up to exponentially small errors that are negligible compared with the estimate of the tunneling [6, Prop. 2.7].

4.3.2 Approximation of ground states

The ground states of ${\mathscr {N}}_{\hbar }^{(1)}$ concentrate near the curvature well $s_1=0$ [6, Corol.6.1] and one can expand them in a WKB form.

Let $\phi _{\hbar ,1}$ be a normalized ground state of ${\mathscr {N}}_{\hbar }^{(1)}$. We introduce the function

$$\begin{aligned} \Phi _1(\sigma ) = \sqrt{\frac{2C_1}{(\mu _1^N)''(\xi _0)}}\int _{[0,\sigma ]}\sqrt{k_{\max }-k(\varsigma )}\mathop {}\!{\textrm{d}}\varsigma , \end{aligned}$$

(4.16)

where $[0,\sigma ]$ is the segment joining 0 to $\sigma$ oriented counter-clockwise; in particular $\int _{[0,\sigma ]}=\int _{\sigma }^0$ if $\sigma < 0$ and $\int _{[0,\sigma ]}=\int _0^{\sigma }$ if $\sigma > 0$. The ground state $\phi _{\hbar ,1}$ is approximated as follows. Let $K\subset \bigl (-\frac{L}{2n}+\eta ,\frac{L}{2n}-\eta \bigr )$ be a compact interval, then we have [6, Prop. 6.3  & Eq. (2.5)]

$$\begin{aligned} \Bigl \Vert \langle \tau \rangle \mathrm e^{\Phi _1/\sqrt{\hbar }}\bigl (\phi _{\hbar ,1}-\mathrm e^{-\mathrm i\gamma _0\sigma /\hbar ^2}\psi _{\hbar ,1} \bigr )\Bigr \Vert _{C^1(K;L^2({\mathbb {R}}_+))}={\mathcal {O}}(h^\infty ), \end{aligned}$$

(4.17a)

where $\langle \tau \rangle =(1+|\tau |^2)^{1/2}$ and $\gamma _0$ is introduced in (4.9).

The function $\psi _{\hbar ,1}$ is defined as follows

$$\begin{aligned} \psi _{\hbar ,1} = \chi \Psi _{\hbar ,1} \end{aligned}$$

(4.17b)

where $\chi \in C^{\infty }({\mathbb {R}};[0,1])$ satisfies

$$\begin{aligned} \chi = 1\text { on }\Bigl (-\frac{L}{2n}+2\eta ,\frac{L}{2n}-2\eta \Bigr ),~ \textrm{supp}\chi \subset \Bigl (-\frac{L}{2n}+\eta ,\frac{L}{2n}-\eta \Bigr ) \end{aligned}$$

(4.17c)

and the function $\Psi _{\hbar ,1}$ has the following expansion [6, Thm. 2.8]

$$\begin{aligned} \mathrm e^{\Phi _1(\sigma )/\hbar ^{\frac{1}{2}}} \mathrm e^{-\mathrm i\sigma \xi _0/\hbar }\Psi _{\hbar ,1}(\sigma ,\tau )\underset{\hbar \rightarrow 0}{\sim }\hbar ^{-\frac{1}{8}} \sum _{j\ge 0} b_{j}(\sigma ,\tau ) \hbar ^{\frac{j}{2}}, \end{aligned}$$

(4.17d)

where

$$\begin{aligned} b_{0}(\sigma ,\tau )=f_0(\sigma )u_0(\tau ) \end{aligned}$$

and the sequence of functions $(b_j(\sigma ,\tau ))_{j\ge 1}$ can be constructed by recursion [7, Thm. 5.6].

4.4 Construction of quasi-modes

Now we can introduce the following quasi-modes for ${\mathscr {T}}_\hbar$

$$\begin{aligned} {\tilde{\phi }}_{\hbar ,1} :=\chi \phi _{\hbar ,1}, \quad {\tilde{\phi }}_{\hbar ,2} :={\mathscr {M}}_n{\tilde{\phi }}_{\hbar ,1},\ldots \quad {\tilde{\phi }}_{\hbar ,n} :={\mathscr {M}}_n{\tilde{\phi }}_{\hbar ,n-1}, \end{aligned}$$

(4.18)

where (see (4.8))

$$\begin{aligned} {\mathscr {M}}_nu(\sigma ,\tau ) = u\left( \sigma -\frac{L}{2n},\tau \right) . \end{aligned}$$

(4.19)

Notice that ${\tilde{\phi }}_{\hbar ,i}$ is supported in ${\tilde{\Gamma }}_{\hbar ,\eta }^{(i)}$ defined as follows

$$\begin{aligned} {\tilde{\Gamma }}_{\hbar ,\eta }^{(i)}=\left( (i-2)\frac{2L}{n}+\eta ,i\frac{2L}{n}-\eta \right) \times (0,\varepsilon _0\hbar ^{-1/2}). \end{aligned}$$

This yields quasi-modes for the operator ${\tilde{T}}_h$ in (4.3) obtained from ${\mathscr {T}}_\hbar$ by the change of parameter $\hbar =h^{1/2}$; more precisely, we introduce

$$\begin{aligned} {\tilde{u}}_{h,i} = {\tilde{\phi }}_{\hbar ,i}\qquad (\hbar =h^{1/2}, ~ 1\le i\le n). \end{aligned}$$

(4.20)

Notice that by (4.17a),

$$\begin{aligned} \mathrm e^{\Phi _2(\sigma ,\tau )/\sqrt{\hbar }}\bigl ({\tilde{u}}_{\hbar ,2}(\sigma ,\tau )- \mathrm e^{2\mathrm i\gamma _0 L/n\hbar ^2} \mathrm e^{-\mathrm i\gamma _0\sigma /\hbar ^2}{\tilde{\psi }}_{\hbar ,1}(\sigma -2L/n,\tau )\bigr )={\mathcal {O}}(h^\infty ), \end{aligned}$$

and by (4.2e),

$$\begin{aligned} \begin{aligned} \Phi _2(\sigma )&= \Phi _1\left( \sigma -\frac{2L}{n}\right) \text{ on } {\tilde{\Gamma }}_{\hbar ,\eta }^{(2)}\\ \Phi _2(\sigma )+\Phi _1(\sigma )&= \sqrt{\frac{2C_1}{(\mu _1^N)''(\xi _0)}} \int _{\sigma -\frac{2L}{n}}^\sigma \sqrt{k_{\max }-k(\varsigma )}\mathop {}\!{\textrm{d}}\varsigma \\&= {\textsf{S}}_n \text{ on } {\tilde{\Gamma }}_{\hbar ,\eta }^{(2)}\cap {\tilde{\Gamma }}_{\hbar ,\eta }^{(1)} \end{aligned} \end{aligned}$$

where ${\textsf{S}}_n$ is introduced in (4.4b).

To obtain quasi-modes for the operator $T_h$ in (4.3), we truncate, re-scale and pull back the quasi-modes ${\tilde{u}}_{h,i}$ to the Cartesian coordinates via the transformation in (4.6) and finally renormalize. More precisely, we introduce

$$\begin{aligned} u_{h,i}(x) = h^{-1/4}\chi _0\big (t(x)/h^{1/2}\big ) {\tilde{u}}_{h,i}\big (s(x),h^{-1/2}t(x)\big ) \end{aligned}$$

(4.21)

where $\chi _0\in C_c^\infty ({\mathbb {R}};[0,1])$ satisfies $\textrm{supp}\chi _0\subset (-\varepsilon _0,\varepsilon _0)$ and $\chi _0=1$ on $[-\varepsilon _0/2,\varepsilon _0/2]$.

4.5 Estimates of interaction coefficients

We need to estimate

$$\begin{aligned} J_0(h)=\langle T_h u_{h,1},u_{h,1}\rangle \text{ and } J_1(h)=\langle T_hu_{h,1}, u_{h,2}\rangle . \end{aligned}$$

By the exponential decay of $u_{h,1}$ and $u_{h,2}$, we may write

$$\begin{aligned} {\tilde{J}}_0(h) = h{\tilde{J}}_0+{\mathcal {O}}(\mathrm e^{-K'/h^{1/4}}), \quad J_1(h) = h{\tilde{J}}_1+{\mathcal {O}}(\mathrm e^{-K'/h^{1/4}}) \end{aligned}$$

where $K'>{\textsf{S}}_n$ and

$$\begin{aligned} {\tilde{J}}_0 = \langle {\mathscr {T}}_\hbar {\tilde{\phi }}_{\hbar ,1},{\tilde{\phi }}_{\hbar ,1}\rangle ,\quad {\tilde{J}}_1 = \langle {\mathscr {T}}_\hbar {\tilde{\phi }}_{\hbar ,1},{\tilde{\phi }}_{\hbar ,2}\rangle \end{aligned}$$

The term ${\tilde{J}}_0$ is estimated as ${\mathcal {O}}(\mathrm e^{(-2{\textsf{S}}_n+c\eta )/\hbar ^{1/2}})$ where c is a positive constant independent of $\hbar$ and $\eta$. We fix now the choice of $\eta \ll 1$ so that $2{\textsf{S}}_n-c\eta >{\textsf{S}}_n$.

The term ${\tilde{J}}_1$ is calculated as in [6, Sec. 7.2.1] (see also [8, 26])

$$\begin{aligned} {\tilde{J}}_1= \mathrm e^{2\mathrm i(\gamma _0-\xi _0)L/n\hbar } \mathrm e^{-{\textsf{S}}_n/\hbar ^{1/2}}\bigl (\hbar ^{5/4} C_* +{\mathcal {O}}(\hbar ^{7/4})\bigr ) \end{aligned}$$

where $C_*\in {\mathbb {C}}\setminus \{0\}$ is a constant independent of $\hbar$.

Writing $C_*=|C_*|\mathrm e^{\mathrm i\alpha _0}$ and recalling the definition of $\Phi _0$ in (4.4a) and the relation $\hbar =h^{1/2}$, we get

$$\begin{aligned} J_1(h)\underset{h\searrow 0}{\sim }\ |C_*|h^{13/8}\mathrm e^{\mathrm i\alpha _0 + 2\mathrm i\Phi _0 /n h^{1/2}}\mathrm e^{-{\textsf{S}}_n/h^{1/4}} \end{aligned}$$

(4.22a)

and

$$\begin{aligned} J_0(h)\underset{h\searrow 0}{=}o\big (J_1(h)\big ). \end{aligned}$$

(4.22b)

4.6 Application when $n=3$.

Let us assume that $n=3$. The functions $u_{h,i}$ introduced in (4.21) satisfy the conditions in Assumptions 2.1, 2.8 (in Remark 2.7) and 2.11. In fact,

• By (4.18) and (4.20), we check that the symmetry invariance in Assumption 2.8 is respected.

• The symmetry invariance and (4.17a) ensure that $u_{h,i}$ satisfy the conditions in Assumption 2.1 with ${\mathfrak {S}}_1,{\mathfrak {S}}_2,{\mathfrak {S}}_3\in (0,{\textsf{S}}_n)$ arbitrarily close to ${\textsf{S}}_n$; we fix the choice so that ${\textsf{S}}_n<2\min _{1\le j\le 3}{\mathfrak {S}}_j$.

• The estimates in (4.22) ensure that Assumption 2.11 holds with ${\mathfrak {S}}={\textsf{S}}_n$.

Applying (2.21), we get a similar result to Theorem 3.9 (by following exactly the same argument). In fact, there is a relabeling $\mu _1(h),\mu _2(h),\mu _3(h)$ of the eigenvalues $\lambda _1(h),\lambda _2(h),\lambda _3(h)$ of the Neumann magnetic Laplacian ${\mathcal {L}}_h^N$ with the asymptotics

$$\begin{aligned} \begin{aligned} \mu _2(h)-\mu _1(h)&\underset{h\searrow 0}{=} \bigl ({\textsf{a}}(\Phi _0/3\sqrt{h}+\alpha _0)+o(1)\bigr ) |C_*|\exp \left( \frac{- {\textsf{S}}_3}{h^{1/2}}\right) \\ \mu _3(h)-\mu _2(h)&\underset{h\searrow 0}{=}\bigl ({\textsf{b}}(\Phi _0/3\sqrt{h}+\alpha _0) +o(1)\bigr ) |C_*|\exp \left( \frac{- {\textsf{S}}_3}{h^{1/2}}\right) \end{aligned} \end{aligned}$$

(4.23)

where ${\textsf{a}}(\cdot )$ and ${\textsf{b}}(\cdot )$ are introduced (3.43).

Moreover, there exists a sequence $\bigl ((h_1(k),h_2(k),h_3(k)\bigr )_{k\ge k_0}$ which converges to 0 such that, for all $k\ge k_0$ we have

$$\begin{aligned} 0<h_1(k+1)<h_3(k)<h_2(k)<h_1(k)<1 \end{aligned}$$

and

$$\begin{aligned} \mu _1\big (h_1(k)\big )=\mu _3\big (h_1(k)\big ),~ \mu _1\big (h_2(k)\big ) = \mu _2\big (h_2(k)\big ),~ \mu _2\big (h_3(k)\big )=\mu _3\big (h_3(k)\big ). \end{aligned}$$

In particular, this finishes the proof of Theorem 1.5.

5 Magnetic steps

5.1 Introduction

In this section, we work in the plane and the Hilbert space is $H=L^2({\mathbb {R}}^2)$. Consider a positive integer n and denote by $g=g_n$ the rotation in ${\mathbb {R}}^2$ by $2\pi /n$. We are then in the setting of Remark 2.7 with the domain being all of ${\mathbb {R}}^2$. Recall the definition of the transformation

$$\begin{aligned} M(g)u(x)=u(g^{-1}x)\quad (u\in L^2({\mathbb {R}}^2)). \end{aligned}$$

Let $\Omega \subset {\mathbb {R}}^2$ be an open bounded subset of ${\mathbb {R}}^2$ with $C^\infty$ boundary $\Gamma$. We assume that $\Omega$ satisfies the conditions in (4.2).

Let ${\textbf{A}}:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2$ be a vector field such that

$$\begin{aligned} B:={{\,\textrm{curl}\,}}{\textbf{A}}= {\left\{ \begin{array}{ll} 1&{}\mathrm{on~}\Omega \\ \vartheta &{}\mathrm{on~}{\mathbb {R}}^2\setminus \overline{\Omega } \end{array}\right. } \end{aligned}$$

(5.1)

where $-1<\vartheta <0$ is a fixed constant. The magnetic field $B=B_{\vartheta ,\Omega }$ is a step function, hence called a magnetic step ([3] and references therein). The boundary $\Gamma$ is the discontinuity curve of the magnetic step, and sometimes is called the magnetic edge ([13] and references therein).

Consider the Landau Hamiltonian on ${\mathbb {R}}^2$

$$\begin{aligned} {\mathscr {L}}_h^B = (-\mathrm ih\nabla -{\textbf{A}})^2 \end{aligned}$$

(5.2)

with semi-classical parameter h and magnetic field B as in (5.1). The symmetry conditions in (4.2) allow us to prove that the eigenvalues of ${\mathscr {L}}_h^B$ exhibit a braid structure in the semi-classical limit. The proof and the construction are very similar to those for the Neumann problem in Section 4.1.2 modulo the following slight modifications:

• i) Use the model operator on the full real line from [2].

• ii) Introduce adapted coordinates on a curved strip defined by the boundary $\Gamma$, via the signed normal distance to $\Gamma$ and the tangential arc-length distance along $\Gamma$.

• iii) Use a modified single well operator, with magnetic steps, analyzed in [3].

• iv) Construct quasi-modes and apply the abstract results in Section 2.

Most of the computations were carried out in [3, 13] so we will be rather succinct and just present the key constructions.

5.2 Model on the real line

On $L^2({\mathbb {R}})$, consider the family of Schrödinger operators, parameterized by $\xi \in {\mathbb {R}}$,

$$\begin{aligned} {\mathfrak {h}}_{\vartheta }[\xi ] = -\frac{d^2}{d\tau ^2}+V_{\vartheta }(\xi ,\tau ), \end{aligned}$$

(5.3)

where $\xi \in {\mathbb {R}}$ is a parameter and

$$\begin{aligned} V_{\vartheta }(\xi ,\tau ) = \big (\xi +b_{\vartheta }(\tau )\tau \big )^2,\quad b_{\vartheta }(\tau ) = \textbf{1}_{{\mathbb {R}}_+}(\tau )+\vartheta \textbf{1}_{{\mathbb {R}}_-}(\tau )\,. \end{aligned}$$

(5.4)

We denote by $\mu _{\vartheta }(\xi )$ the ground state energy of ${\mathfrak {h}}_{\vartheta }[\xi ]$, and according to [2], we can introduce the constants $\beta _{\varsigma },\zeta _{\vartheta }$ as follows

$$\begin{aligned} \beta _{\vartheta } :=\inf _{\xi \in {\mathbb {R}}} \mu _{\vartheta }(\xi )=\mu _{\vartheta }(\zeta _{\vartheta })\,, \end{aligned}$$

(5.5)

where $\zeta _{\vartheta }<0$, is the unique minimum of $\mu _{\vartheta }(\cdot )$ and we have $\mu _{\vartheta }''(\zeta _{\vartheta })>0$.

Let $\phi _{\vartheta }$ be the positive $L^2$-normalized ground state of ${\mathfrak {h}}_\vartheta [\zeta _{\vartheta }]$. We introduce the negative constant [2]

$$\begin{aligned} M_3(\vartheta ) = \frac{1}{3}\Big (\frac{1}{\vartheta }-1\Big )\zeta _{\vartheta }\phi _{\vartheta }(0)\phi _{\vartheta }'(0). \end{aligned}$$

(5.6)

5.3 Adapted coordinates and single well problem

Recall that $\gamma :{\mathbb {R}}/2L{\mathbb {Z}}\rightarrow {\mathbb {R}}^2$ is a counter-clockwise oriented arc-length parameterization of the curve $\Gamma$ and that, for $s\in {\mathbb {R}}/2L{\mathbb {Z}}$, $\nu (s)$ is the unit normal to $\Gamma$ at $\gamma (s)$ which points to the interior of the $\Gamma$. We can adjust the coordinates (s, t) introduced in (4.6), by allowing t to have negative values. We therefore set

$$\begin{aligned} {\mathscr {T}}(s,t):=\gamma (s)-t\nu (s)\qquad ((s,t)\in {\mathbb {R}}/2L{\mathbb {Z}}\times {\mathbb {R}}). \end{aligned}$$

We introduce the curved strip

$$\begin{aligned} {\hat{\Gamma }}(\varepsilon _0)=\{x\in {\mathbb {R}}^2~:~\textrm{dist}(x,\Gamma )<\varepsilon _0\} \end{aligned}$$

and we choose $\varepsilon _0>0$ so that ${\mathscr {T}}$ is one-to-one on ${\mathbb {R}}/2L{\mathbb {Z}}\times (-\varepsilon _0,\varepsilon _0)$ and that ${\hat{\Gamma }}(\varepsilon _0)$ is the image of ${\mathbb {R}}/2L{\mathbb {Z}}\times (-\varepsilon _0,\varepsilon _0)$ by ${\mathscr {T}}$.

We assign to a function u defined on ${\hat{\Gamma }}(\varepsilon _0)$ the function ${\tilde{u}}=u\circ {\mathscr {T}}$ defined on ${\mathbb {R}}/2L{\mathbb {Z}}\times (-\varepsilon _0,\varepsilon _0)$. The Hilbert space $L^2({\hat{\Gamma }}(\varepsilon _0))$ is then transformed to the weighted space

$$\begin{aligned} L^2\big (({\mathbb {R}}/ 2L {\mathbb {Z}}) \times (-\varepsilon _0,\varepsilon _0); a\mathop {}\!{\textrm{d}}s\mathop {}\!{\textrm{d}}t\big ),\quad a(s,t)=1-tk(s) \end{aligned}$$

and the action of the transformation M(g) is still given by (4.8).

Note that the action of ${\mathscr {L}}_h^B$ on ${\hat{\Gamma }}(\varepsilon _0)$ is transformed to (after, possibly, a gauge transformation)

$$\begin{aligned} \tilde{{\mathscr {L}}}_h^B{} & {} :=-h^2 a^{-1}\partial _t a\partial _t + a^{-1}\left( -\mathrm ih\partial _s+\gamma _0-b_{\vartheta }(t)t+\frac{k(s)}{2}b_{\vartheta }(t)t^2\right) \\{} & {} \qquad \times a^{-1}\left( -\mathrm ih\partial _s+\gamma _0-b_{\vartheta }(t)t+\frac{k(s)}{2}b_{\vartheta }(t)t^2\right) \end{aligned}$$

where the constant $\gamma _0$ and the function $b_{\vartheta }$ are introduced in (4.9) and (5.4) respectively.

We do the scaling $(\sigma ,\tau )=(s,h^{-1/2}t)$ and introduce the effective semi-classical parameter $\hbar =h^{1/2}$. We obtain the operator

$$\begin{aligned} {\mathscr {N}}_{\hbar ,\vartheta }= & {} - a^{-1}_{\hbar }\partial _\tau a_{\hbar }\partial _\tau + {a}^{-1}_{\hbar }\left( -\mathrm i\hbar \partial _\sigma +{ \hbar ^{-1}\gamma _0}-b_{\vartheta }\tau +\hbar \frac{k(\sigma )}{2}b_{\vartheta }\tau ^2\right) \nonumber \\{} & {} \times {a}^{-1}_{\hbar }\left( -\mathrm i\hbar \partial _\sigma +{ \hbar ^{-1}\gamma _0}-b_{\vartheta }\tau +\hbar \frac{k(\sigma )}{2}b_{\vartheta }\tau ^2\right) \end{aligned}$$

(5.7)

where¹¹

$$\begin{aligned} a_\hbar (\sigma ,\tau )=1-\hbar \tau k(\sigma ). \end{aligned}$$

Our single well problem is defined as follows. Fix a positive $\eta <\min (\frac{1}{4},\frac{L}{4n})$ and consider

$$\begin{aligned} {\hat{\Gamma }}_{\hbar ,\eta }^{(1)}=\Bigl (-\frac{2L}{n}+\eta ,\frac{2L}{n}-\eta \Bigr )\times (-\varepsilon _0\hbar ^{-1},\varepsilon _0\hbar ^{-1}). \end{aligned}$$

We introduce the operator on $L^2\big ({\hat{\Gamma }}_{\hbar ,\eta }^{(1)}; a_{\hbar }\mathop {}\!{\textrm{d}}\sigma \mathop {}\!{\textrm{d}}\tau \big )$,

$$\begin{aligned} {\mathscr {N}}_{\hbar ,\vartheta }^{(1)}={\mathscr {N}}_{\hbar ,\vartheta } \end{aligned}$$

(5.8)

with domain

$$\begin{aligned} \begin{aligned} \textsf{Dom}({\mathscr {N}}_{\hbar ,\vartheta }^{(1)})=\{u\in L^2\big ({\hat{\Gamma }}_{\hbar ,\eta }^{(1)};\mathop {}\!{\textrm{d}}\sigma \mathop {}\!{\textrm{d}}\tau \big )~:~&\partial _\tau ^2u,\partial _\sigma ^2u\in L^2\big ({\hat{\Gamma }}_{\hbar ,\eta }^{(1)};\mathop {}\!{\textrm{d}}\sigma \mathop {}\!{\textrm{d}}\tau \big ),\\&u|_{\tau =\pm \varepsilon _0h^{-1/2}}=0,~ u|_{\sigma =\pm (2L/n-\eta )}=0\}. \end{aligned} \end{aligned}$$

We denote by $\lambda (\hbar )$ the ground state energy of the operator ${\mathscr {N}}_{\hbar ,\vartheta }^{(1)}$; it is simple and can be expanded as follows [3, 13]

$$\begin{aligned} \lambda (\hbar )\underset{\hbar \rightarrow 0}{\sim }\beta _{\varsigma }+M_3(\vartheta )k_{\max }\hbar +\sqrt{\frac{k_2M_3(\vartheta )\mu _{\vartheta }''(\zeta _{\vartheta })}{4} }\hbar ^{3/2}, \end{aligned}$$

(5.9)

and the splitting between $\lambda (\hbar )$ and the second eigenvalue $\lambda _2(\hbar )$ is estimated as follows

$$\begin{aligned} \lambda _2(\hbar )-\lambda (\hbar )\underset{\hbar \rightarrow 0}{\sim }\sqrt{\frac{k_2M_3(\vartheta )\mu _{\vartheta }''(\zeta _{\vartheta })}{4} }\hbar ^{3/2}. \end{aligned}$$

5.4 Quasi-modes and application

In order to apply the results in Section 2 we introduce the operator

$$\begin{aligned} {\mathcal {T}}_h={\mathscr {L}}_h^B-\ell (h) \end{aligned}$$

and construct suitable quasi-modes. With $\hbar =h^{1/2}$, we choose $\ell (h)=h\lambda (\hbar )$, where $\lambda (\hbar )$ is the ground state energy of the single well operator ${\mathscr {N}}_{\hbar ,\vartheta }^{(1)}$ introduced in (5.8).

Let $\phi _{\hbar ,1}^\vartheta$ be a normalized ground state of ${\mathscr {N}}_{\hbar ,\vartheta }^{(1)}$ and choose $\chi \in C^{\infty }({\mathbb {R}};[0,1])$ satisfying (4.17c). We introduce the quasi-modes

$$\begin{aligned} {\tilde{\phi }}_{\hbar ,1}^\vartheta :=\chi \phi _{\hbar ,1}^\vartheta ,\quad {\tilde{\phi }}_{\hbar ,2}^\vartheta :={\mathscr {M}}_n{\tilde{\phi }}_{\hbar ,1}^\vartheta ,\ldots \quad {\tilde{\phi }}_{\hbar ,n}^\vartheta :={\mathscr {M}}_n{\tilde{\phi }}_{\hbar ,n-1}^\vartheta , \end{aligned}$$

(5.10)

where ${\mathscr {M}}_n$ is introduced in (4.19). We obtain quasi-modes for the operator $T_h$ by truncation, re-scaling and pulling back to Cartesian coordinates; more precisely we introduce

$$\begin{aligned} u_{h,i}^\vartheta (x)= h^{-1/4}\chi _0\big (t(x)/h^{1/2}\big ) {\tilde{u}}_{h,i}^\vartheta \big (s(x),h^{-1/2}t(x)\big ) \end{aligned}$$

(5.11)

where $\chi _0\in C_c^\infty ({\mathbb {R}};[0,1])$ is as in (4.21) and ${\tilde{u}}_{h,i}^\vartheta ={\tilde{\phi }}_{\hbar ,i}^\vartheta$, with $\hbar =h^{1/2}$.

We move then to the calculation of the following two terms

$$\begin{aligned} J_0(h)=\langle {\mathcal {T}}_h u_{h,1}^\vartheta ,u_{h,1}^\vartheta \rangle \text{ and } J_1(h)=\langle {\mathcal {T}}_hu_{h,1}^\vartheta , u_{h,2}^\vartheta \rangle . \end{aligned}$$

That is essentially done in [13]. We introduce the following ‘distance’

$$\begin{aligned} {\textsf{S}}_n(\vartheta )=\sqrt{\frac{ -2M_3(\vartheta )}{\mu _{\vartheta }''(\zeta _{\vartheta })}}\int _0^{2L/n}\sqrt{k_{\max } -k(s)}\mathop {}\!{\textrm{d}}s \end{aligned}$$

and the flux like term

$$\begin{aligned} \Phi _\vartheta =|\Omega |-2L\zeta _{\vartheta }. \end{aligned}$$

We have

$$\begin{aligned} J_0(h)= {\mathcal {O}}(\mathrm e^{(-2{\textsf{S}}_n(\vartheta )+c\eta )/h^{1/4}}) \end{aligned}$$

where c is a positive constant independent of $\hbar$ and $\eta$, and

$$\begin{aligned} {\tilde{J}}_1(h)= \mathrm e^{2\mathrm i\Phi _\vartheta /n\hbar } \mathrm e^{-{\textsf{S}}_n(\vartheta )/h^{1/4}}\bigl ( C_*(\vartheta )h^{13/8} +{\mathcal {O}}(h^{15/8})\bigr ) \end{aligned}$$

where $C_*(\vartheta )\in {\mathbb {C}}\setminus \{0\}$ is a constant independent of $\hbar$.

We fix now the choice of $\eta \ll 1$ so that $2{\textsf{S}}_n(\vartheta )-c\eta >{\textsf{S}}_n(\vartheta )$ and we write $C_*(\vartheta )=|C_*(\vartheta )|\mathrm e^{\mathrm i\alpha _0(\vartheta )}$. We get

$$\begin{aligned} J_1(h)\underset{h\searrow 0}{\sim } |C_*(\vartheta )|h^{13/8}\mathrm e^{\mathrm i\alpha _0 (\vartheta ) + 2\mathrm i\Phi _\vartheta /n h^{1/2}}\mathrm e^{-{\textsf{S}}_n(\vartheta )/h^{1/4}} \end{aligned}$$

(5.12a)

and

$$\begin{aligned} J_0(h)\underset{h\searrow 0}{=}o\big (J_1(h)\big ). \end{aligned}$$

(5.12b)

Let us now assume that $n=3$, which corresponds to the setting in Theorem 1.6. The functions $u_{h,i}^\vartheta$ introduced in (4.21) satisfy the conditions in Assumptions 2.1, 2.8 (in Remark 2.7) and 2.11. Applying (2.21), we get a relabeling $\mu _1(h),\mu _2(h),\mu _3(h)$ of the eigenvalues $\lambda _1(h),\lambda _2(h),\lambda _3(h)$ of the Landau hamiltonian ${\mathscr {L}}_h^B$ with the asymptotics

$$\begin{aligned} \begin{aligned} \mu _2(h)-\mu _1(h)&\underset{h\searrow 0}{=} \bigl ({\textsf{a}}(\Phi _\vartheta /3\sqrt{h}+\alpha _0(\vartheta ))+o(1)\bigr ) |C_*(\vartheta )|\exp \left( \frac{- {\textsf{S}}_3(\vartheta )}{h^{1/2}}\right) \\ \mu _3(h)-\mu _2(h)&\underset{h\searrow 0}{=}\bigl ({\textsf{b}}(\Phi _\vartheta /3\sqrt{h}+\alpha _0(\vartheta )) +o(1)\bigr ) |C_*(\vartheta )|\exp \left( \frac{- {\textsf{S}}_3(\vartheta )}{h^{1/2}}\right) \end{aligned} \end{aligned}$$

where ${\textsf{a}}(\cdot )$ and ${\textsf{b}}(\cdot )$ are introduced (3.43).

Moreover, there exists a sequence $\bigl ((h_1(k),h_2(k),h_3(k)\bigr )_{k\ge k_0}$ which converges to 0 such that, for all $k\ge k_0$ we have

$$\begin{aligned} 0<h_1(k+1)<h_3(k)<h_2(k)<h_1(k)<1 \end{aligned}$$

and

$$\begin{aligned} \mu _1\big (h_1(k)\big )=\mu _3\big (h_1(k)\big ),~ \mu _1\big (h_2(k)\big ) = \mu _2\big (h_2(k)\big ),~ \mu _2\big (h_3(k)\big )=\mu _3\big (h_3(k)\big ). \end{aligned}$$

In particular, this finishes the proof of Theorem 1.6.

Minimizing the function \(\Psi (r,t)\).

Let $\Psi$ be the function introduced in (3.19). Using the inequality $\sqrt{t(t+1)}\le t+\frac{1}{2}$ we get

$$\begin{aligned} \Psi (r,t)\ge d(r)+\frac{|v_0^{\min }|}{2}\ln \Bigl (1+\frac{1}{t}\Bigr )+\frac{(L-r)^2}{2}\left( t+\frac{1}{2}\right) . \end{aligned}$$

(A.1)

Consequently, the minimum of $\Psi (r,t)$ over $[a,L-a]\times {\overline{{\mathbb {R}}}}_+$ is attained at $(r_0,t_0)\in [a,L-a]\times {\mathbb {R}}_+$. By [17, Remark 4.8],

$$\begin{aligned} \inf _{(r,t)\in [a,L-a]\times {\mathbb {R}}_+}{\Psi (r,t)} \ge \inf \left\{ \inf _{(r,t)\in [0,a]\times {\mathbb {R}}_+}\Psi (r,t),\inf _{t\in {\mathbb {R}}_+}\Psi (L-a,t) \right\} . \end{aligned}$$

(A.2)

Recalling $F(v_0)$ from (3.19), we have by [17, Eq. (4.18) and Prop. 4.5],

$$\begin{aligned} -F(v_0)+\inf _{(r,t)\in [0,a]\times {\mathbb {R}}_+}\Psi (r,t)= S(v_0,L)\ge S_a \end{aligned}$$

(A.3)

where $S(v_0,L) \text{ and } S_a$ are introduced in (3.9).

The function

$$\begin{aligned} G(t) := \frac{|v_0^{\min }|}{2}\ln \Bigl (1+\frac{1}{t}\Bigr )+\frac{a^2}{2}\left( t+\frac{1}{2}\right) \end{aligned}$$

has a unique minimum on ${\mathbb {R}}_+$,

$$\begin{aligned} t_* = \sqrt{\frac{1}{4}+\frac{|v_0^{\min }|}{a^2}}-\frac{1}{2}. \end{aligned}$$

By (A.1) (for $r=L-a$), we get

$$\begin{aligned} \begin{aligned} \inf _{t\in {\mathbb {R}}_+}\Psi (L-a,t)&\ge d(L-a)+G(t_*)\\&=d(L-a)+\frac{a}{4}\sqrt{a^2+4|v_0^{\min }|}+\frac{|v_0^{\min }|}{2}\ln \frac{ \left( \sqrt{a^2+4|v_0^{\min }|} +a\right) ^2}{4|v_0^{\min }|}. \end{aligned} \end{aligned}$$

Consequently,

$$\begin{aligned} -F(v_0)+\inf _{t\in {\mathbb {R}}_+}\Psi (L-a,t)\ge d(L-a)+d(a)=S_a. \end{aligned}$$

(A.4)

Collecting (A.3) and (A.4), we infer from (A.2) that

$$\begin{aligned} -F(v_0)+\inf _{(r,t)\in [a,L-a]\times {\mathbb {R}}_+}\Psi (r,t)\ge S_a. \end{aligned}$$