Introduction

The main topic of this paper is the one-dimensional massless Dirac Hamiltonian with a two-parameter perturbation proportional to the Coulomb potential

$$\begin{aligned} D_{\omega , \lambda } = \begin{bmatrix} -\frac{\lambda +\omega }{x} &{} - \partial _x \\ \partial _x &{} -\frac{\lambda -\omega }{x} \end{bmatrix}. \end{aligned}$$
(1.1)

We allow the parameters \(\omega ,\lambda \) to be complex. We will describe realizations of (1.1) as closed operators on \(L^2(\mathbb {R}_+,\mathbb {C}^2).\) We will call (1.1) the one-dimensional Dirac–Coulomb Hamiltonian or operator (omitting usually the adjective one-dimensional, or shortening it to 1d).

The formal operator \( D_{\omega , \lambda } \) is homogeneous of degree \(-1\). Among its various closed realizations we will be especially interested in homogeneous ones, i.e., those whose domain is invariant with respect to scaling transformations.

Our main motivation to study \(D_{\omega , \lambda }\) comes from the 3d Dirac–Coulomb Hamiltonian

$$\begin{aligned} \sum _{j=1}^3 \alpha _j p_j +\beta m-\frac{\lambda }{r}\end{aligned}$$
(1.2)

acting on four component spinor functions on \(\mathbb {R}^3\). Here \(m \in \mathbb {R}\) is the mass parameter, \(\lambda \in \mathbb {R}\) is related to the charge of nucleus and \(p_j:=-\text {i}\partial _{x^j}\). As is well known, after separation of variables in (1.2) with \(m=0\) one obtains (1.1). Possible values of \(\omega \) are \(\pm 1, \pm 2,\dots \). They are related to the angular momentum. Similar separation is possible also in other dimensions, albeit leading to different values of \(\omega \). We remark that the mass term is bounded and hence does not change the domain. Therefore, the analysis of the \(m=0\) case yields the description of closed realizations of the massive Dirac–Coulomb operator.

The second source of interest in \(D_{\omega , \lambda }\) is the expectation that models with scaling symmetry describe the behavior of much more complicated systems in certain limiting cases.

There exists another important motivation for the study of Dirac–Coulomb Hamiltonians. Objects related to (1.1), such as its eigenfunctions and Green’s kernels can be expressed in terms of Whittaker functions (or, equivalently, confluent functions). Whittaker functions are eigenfunctions of the Whittaker operator

$$\begin{aligned} L_{\beta ,\alpha } :=-\partial _x^2+\Big (\alpha -\frac{1}{4}\Big )\frac{1}{x^2}-\frac{\beta }{x}. \end{aligned}$$
(1.3)

The Dirac–Coulomb Hamiltonian may be viewed as a good way to organize our knowledge about Whittaker functions, one of the most important families of special functions in mathematics. Curiously, it seem more suitable for this goal than the Whittaker operator itself. Indeed, the homogeneity of the Dirac–Coulomb operator leads to several identities which have no counterparts in the case of the Whittaker operator (e.g., the scattering theory described in Sect. 6 with [13] and [10]).

Let us briefly describe the content of our paper. The most obvious closed realizations of \( D_{\omega , \lambda } \) are the minimal and maximal realizations, denoted \( D_{\omega , \lambda }^{\min } \) and \( D_{\omega , \lambda }^{\max } \). Both are homogeneous of degree \(-1\). They depend holomorphically on parameters \(\omega ,\lambda \), except for \(|\text {Re}\sqrt{\omega ^2-\lambda ^2}|=\frac{1}{2}\), where a kind of a “phase transition” occurs. One of the signs of this phase transition is the following: For \(|\text {Re}\sqrt{\omega ^2-\lambda ^2}| \ge \frac{1}{2}\), we have \( D_{\omega , \lambda }^{\min } = D_{\omega , \lambda }^{\max }\), so that in this parameter range there is only one closed realization of \( D_{\omega , \lambda } \). However, for \(|\text {Re}\sqrt{\omega ^2-\lambda ^2}|<\frac{1}{2}\), the domain of \( D_{\omega , \lambda }^{\min } \) has codimension 2 as a subspace of the domain of \( D_{\omega , \lambda }^{\max }\). This means that for fixed \((\omega , \lambda )\) in this region there exists a one-parameter family of closed realizations of \(D_{\omega , \lambda } \) strictly between the minimal and maximal realization.

In operator theory (and other domains of mathematics) it is useful to organize objects in holomorphic families [14, 29]. Therefore, we ask whether \( D_{\omega , \lambda }^{\min }= D_{\omega , \lambda }^{\max }\) can be analytically continued beyond the region \(|\text {Re}\sqrt{\omega ^2-\lambda ^2}|>\frac{1}{2}\). The answer is positive, but the domain of this continuation is a complex manifold which is not simply an open subset of the “\((\omega ,\lambda )\)-plane” \(\mathbb {C}^2\). To define this manifold we start with the following subset of \(\mathbb {C}^3\):

$$\begin{aligned} \Big \{(\omega ,\lambda ,\mu )\ |\ \mu ^2=\omega ^2-\lambda ^2,\quad \mu >-\frac{1}{2}\Big \}. \end{aligned}$$
(1.4)

Then we “blow up” the singularity \((\omega ,\lambda ,\mu )=(0,0,0)\). The resulting complex two-dimensional manifold is denoted \(\mathcal {M}_{-\frac{1}{2}}\). There exists a natural projection \(\mathcal {M}_{- \frac{1}{2}} \rightarrow \mathbb {C}^2\). The preimage of \((\omega , \lambda ) \in \mathbb {C}^2\) has one element if \(|\text {Re}\sqrt{\omega ^2 - \lambda ^2}| \ge \frac{1}{2}\), two elements if \(|\text {Re}\sqrt{\omega ^2 - \lambda ^2}| < \frac{1}{2}\), and \((\omega , \lambda ) \ne (0,0)\) and infinitely many elements if \(\omega = \lambda =0\). This last preimage, called the zero fiber, is isomorphic to the Riemann sphere \(\mathbb {C}\mathbb {P}^1\), for which we use homogeneous coordinates [a : b]. Away from the zero fiber, points of \(\mathcal {M}_{- \frac{1}{2}}\) may be labeled by triples \((\omega , \lambda , \mu )\).

The main result of our paper is the construction of a holomorphic family of closed operators \(\mathcal {M}_{-\frac{1}{2}}\ni p\mapsto D_p\) consisting of homogeneous Dirac–Coulomb Hamiltonians. If \(p\in \mathcal {M}_{-\frac{1}{2}}\) lies over \((\omega ,\lambda )\), then we have inclusions

$$\begin{aligned} \text {Dom}( D_{\omega , \lambda }^{\min })\subset \text {Dom}( D_p)\subset \text {Dom}( D_{\omega , \lambda }^{\max }). \end{aligned}$$
(1.5)

If \(|\text {Re}\sqrt{\omega ^2 - \lambda ^2}| \ge \frac{1}{2}\), both inclusions in (1.5) are equalities. On the other hand, for \(|\text {Re}\sqrt{\omega ^2 - \lambda ^2}| < \frac{1}{2}\) both inclusions are proper and elements of the domain of \( \text {Dom}(D_p)\) are distinguished by the following behavior near zero:

$$\begin{aligned} \sim \frac{x^{\mu }}{\omega + \lambda } \begin{bmatrix}-\mu \\ \omega +\lambda \end{bmatrix},\qquad \sim \frac{x^{\mu }}{\omega - \lambda }\begin{bmatrix}\omega -\lambda \\ -\mu \end{bmatrix}. \end{aligned}$$
(1.6)

Note that the two functions in (1.6), when both well defined, are proportional to one another.

We describe various properties of \(D_p\): we find its point spectrum, essential spectrum, numerical range, discuss conditions for (maximal) dissipativity. We construct explicitly the resolvent. Some spectral properties, including their point spectra, of nonhomogeneous realizations of \(D_{\omega ,\lambda }\) are also discussed.

Whenever \(D_p\) is self-adjoint, its spectrum is absolutely continuous, simple and coincides with \(\mathbb {R}\). In non-self-adjoint cases, the essential spectrum is still \(\mathbb {R}\), but on certain exceptional subsets of the parameter space there is also point spectrum \(\{\text {Im}(k) > 0\}\) or \(\{\text {Im}(k) < 0\}\). Away from exceptional sets \(D_p\) possesses non-square-integrable eigenfunctions, which can be called distorted waves. They can be normalized in two ways: as incoming and outgoing distorted waves. They define the integral kernels of a pair of operators \(\mathcal {U}^\pm \) that, at least formally, diagonalize \(D_p\). More precisely, on a dense domain \(\mathcal {U}^\pm \) intertwine \(D_p\) with the operator of the multiplication by the independent variable \(k\in \mathbb {R}\). Up to a trivial factor, \(\mathcal {U}^\pm \) can be interpreted as the wave (Møller) operators. The operators \(\mathcal {U}^+\) and \(\mathcal {U}^-\) are related to one another by the identity \(S \mathcal {U}^-:= \mathcal {U}^+\), which defines the scattering operator S. Thus, we are able to describe rather completely the stationary scattering theory of homogeneous Dirac–Coulomb Hamiltonians.

For self-adjoint \(D_p\), the operators \(\mathcal {U}^\pm \) are unitary. If \(\lambda \) is real, they are still bounded and invertible, even if \(D_p\) are not self-adjoint. We show that \(\mathcal {U}^\pm \) can be written (up to a trivial factor) as \(\Xi ^\pm (\text {sgn}(k),A)\), where A is the dilation generator and \(\text {sgn}(k)\) is the sign of the spectral parameter. We express \(\Xi ^\pm \) in terms of the hypergeometric function. We prove that they behave as \(s^{|\text {Im}(\lambda )|}\) for \(s\rightarrow \infty \). In particular, this shows that \(\mathcal {U}^\pm \) are bounded only for real \(\lambda \).

The Coulomb potential is long-range. Therefore, we cannot use the standard formalism of scattering theory. In our paper we restrict ourselves to the stationary formalism, where the long-range character of the perturbation is taken into account by using appropriately modified plane waves.

Operators \(D_p\) with p in the zero fiber can be fully analyzed by elementary means. All operators strictly between \(D_{0,0}^{\min }\) and \(D_{0,0}^{\max }\) are homogeneous and are specified by boundary conditions at zero of the form \(f(0) \in \mathbb {C}\begin{bmatrix} a \\ b \end{bmatrix}\) for \([a{:}b]\in \mathbb {C}\mathbb {P}^1\). Operator corresponding to boundary condition [a : b] will be denoted \(D_{[a:b]}\). Other cases in which operators \(D_p\) are particularly simple are discussed in “Appendix A”.

Fig. 1
figure 1

Phase diagram of operators \(D_{\omega , \lambda }\) for \((\omega ,\lambda )\in \mathbb {R}^2\). It is partitioned into five subsets corresponding to five possible behaviors, see Propositions 1 and 2 and Fig. 3. We label regions as follows. Color green and letter A refer to \(\omega ^2 - \lambda ^2 \ge \frac{1}{4}\) (we do not give a separate name to the boundary of this region, although it is also somewhat special). Color blue and letter B refer to the subset \(0< \omega ^2 - \lambda ^2 < \frac{1}{4}\). Black lines and letter C refer to the lines \(\omega = \pm \lambda \), except for the special point \((\omega , \lambda )=(0,0)\), which is marked with a fat red dot and letter D. Yellow color and letter E are used for the region \(\omega ^2 - \lambda ^2 < 0\). In addition we present lines corresponding to the lowest angular momentum values for dimensions \(d=0\), \(d=1\), \(d=2\) and \(d=3\). Here we disregard the possible sign of \(\omega \), which is irrelevant due to symmetry \(\omega \mapsto - \omega \)

Fig. 2
figure 2

Parameter space of homogeneous self-adjoint Dirac–Coulomb Hamiltonians projected onto axes \(\omega , \lambda , \mu \). Regions colored yellow, blue and red are described by inequalities \(\mu > \frac{1}{2}\), \(0< \mu < \frac{1}{2}\) and \(- \frac{1}{2}< \mu < 0\), respectively. The fat dot at the origin represents a circle contained in the zero fiber \(\mathcal {Z}\), so the whole parameter space is topologically a cylinder

The operator \(D_{\omega , \lambda }^{\min }\) is Hermitian (symmetric with respect to the scalar product \((\cdot |\cdot )\)) if and only if \(\omega , \lambda \in {\mathbb {R}}\). Below we state our main results about self-adjoint realizations of \( D_{\omega , \lambda } \) in the form of two propositions. They are immediate consequences of the results of Sects. 4, 5. We present also the phase diagram of operators \(D_{\omega , \lambda }\) on Fig. 1 and the parameter space of homogeneous self-adjoint Dirac–Coulomb Hamiltonians on Fig. 2.

Let \(H^1(\mathbb {R}_+)\) be the first Sobolev space on \(\mathbb {R}_+\) and \(H_0^1(\mathbb {R}_+)\) be the closure of \(C_\mathrm {c}^{\infty }\) in \(H^1(\mathbb {R}_+)\).

Proposition 1

Let \(\omega , \lambda \in {\mathbb {R}}\). The Hermitian operator \(D_{\omega , \lambda }^{\min }\) has the following properties.

  1. 1.

    If \(\frac{1}{4}\le \omega ^2 - \lambda ^2\), it is self-adjoint and \(D_{\omega , \lambda }^{\min }=D_{\omega , \lambda ,\sqrt{\omega ^2-\lambda ^2}}\)

  2. 2.

    If \(\omega ^2 - \lambda ^2<\frac{1}{4}\), it has deficiency indices (1, 1). Hence, there exists a circle of self-adjoint extensions.

Proposition 2

  1. 1a.

    If \(\frac{1}{4}<\omega ^2 - \lambda ^2\), we have \(\text {Dom}(D_{\omega , \lambda }^{\min }) = H_0^1(\mathbb {R}_+,\mathbb {C}^2)\).

  2. 1b.

    If \(\frac{1}{4}=\omega ^2 - \lambda ^2\), we have \(H_0^1(\mathbb {R}_+,\mathbb {C}^2) \subsetneq \text {Dom}(D_{\omega , \lambda }^{\min }) \).

  3. 2a.

    If \(0<\omega ^2 - \lambda ^2<\frac{1}{4}\), exactly two self-adjoint extensions of \(D_{\omega , \lambda }^{\min }\) are homogeneous, namely \(D_{\omega , \lambda , \sqrt{\omega ^2 - \lambda ^2}}\) and \(D_{\omega , \lambda , -\sqrt{\omega ^2 - \lambda ^2}}\). The former is distinguished among all self-adjoint extensions by

    $$\begin{aligned} \int _{0}^{\infty } \left( \frac{|f(x)|^2}{x} + |f(x)| |f'(x)| \right) \text {d}x < \infty \text { for }f \in \text {Dom}(D_{\omega , \lambda , \sqrt{\omega ^2 - \lambda ^2}}),\end{aligned}$$
    (1.7)

    i.e., elements of its domain have finite expectation values of kinetic and potential energy.

  4. 2b.

    If \(| \lambda | = |\omega | \ne 0\), exactly one self-adjoint extension of \(D_{\omega , \lambda }^{\min }\) is homogeneous, namely \(D_{\omega , \lambda , 0}\). It has the property \(H_0^1(\mathbb {R}_+, \mathbb {C}^2) \subsetneq \text {Dom}(D_{\omega , \lambda , 0}) \subsetneq H^{1}(\mathbb {R}_+,\mathbb {C}^2)\).

  5. 2c.

    If \(\lambda = \omega =0\), all self-adjoint extensions of \(D_{\omega , \lambda }^{\min }\) are homogeneous. They have the form \(D_{[a:b]}\) with \([a:b] \in \mathbb {R}\mathbb {P}^1\).

  6. 2d.

    If \(|\lambda | > |\omega |\), none of self-adjoint extensions of \(D_{\omega , \lambda }^{\min }\) is homogeneous.

Now let \(\omega , \lambda \) be real and suppose that \(\omega ^2 - \lambda ^2 < \frac{1}{4}\). Let \(\tau \mapsto U_\tau \) denote the scaling transformation. The parameter space of self-adjoint extensions is a circle. It admits an action of the scaling group given by

$$\begin{aligned} D \mapsto U_{\tau } D U_{\tau }^{-1}. \end{aligned}$$
(1.8)

The fixed points of this action are the homogeneous self-adjoint extensions. Main properties of this action are illustrated by Fig. 3.

Fig. 3
figure 3

Visualization of the action of the scaling group on self-adjoint extensions of \(D_{\omega , \lambda }^{\min }\) in four regions covering the set of \(\omega , \lambda \) satisfying \(\omega ^2 - \lambda ^2 < \frac{1}{4}\). Fat dots are the fixed points, while arrowheads indicate the direction of the flow as \(\tau \) increases, see (1.8). In the first region there are two fixed points, attractive and repulsive, corresponding to a positive and negative \(\mu \), respectively. As \(\omega ^2 - \lambda ^2\) decreases to zero, the two fixed points merge to one degenerate fixed point, except for the point \(\omega = \lambda =0\) at which the scaling action becomes trivial. As \(\omega ^2 - \lambda ^2\) decreases below zero, the scaling action becomes periodic with period \(\frac{\pi }{\sqrt{\lambda ^2 - \omega ^2}}\)

As we present in “Appendix B”, d-dimensional Dirac–Coulomb Hamiltonians can be reduced to the radial operator (1.1). Combined with the analysis presented above, one obtains rather complete information about self-adjointness and homogeneity properties of these operators. Here we point out only a few facts concerning these extensions on the lowest angular momentum sector.

  • dimension 1 There exist no homogeneous self-adjoint realizations for any \(\lambda \ne 0\).

  • dimension 2 The operator defined on smooth spinor-valued functions with compact support not containing zero is not essentially self-adjoint for any \(\lambda \ne 0\). For \(|\lambda |< 1\) there exist homogeneous self-adjoint extensions of \(D_{\omega , \lambda }^{\min }\). These homogeneous extensions can be organized into two continuous families. The (more physical) family is defined on \([-1,1]\). At the endpoints \(\lambda =\pm 1\) it meets the other family, which is defined on \([-1,0[\cup ]0,1]\).

  • dimension \(d \ge 3\): The operator defined as above is essentially self-adjoint if \(\lambda ^2 \le \frac{d(d-2)}{4}\). If \(\frac{d(d-2)}{4} < \lambda ^2\) it is not essentially self-adjoint. However, for \(\lambda ^2\le \frac{(d-1)^2}{4}\) there exists homogeneous self-adjoint extensions of \(D_{\omega , \lambda }^{\min }\). They can be organized into two families depending continuously on \(\lambda \). The more physical family is defined on \([- \frac{(d-1)^2}{4}, \frac{(d-1)^2}{4}]\). The second family meets the first at the endpoints and is defined on \([- \frac{(d-1)^2}{4}, -\frac{d(d-2)}{4}[\,\cup \, ]\frac{d(d-2)}{4}, \frac{(d-1)^2}{4}]. \)

In all cases in which there exist no homogeneous self-adjoint extensions, the defect indices are nevertheless equal and hence there exist nonhomogeneous self-adjoint extensions.

Analysis of self-adjoint realizations of the three-dimensional Dirac–Coulomb Hamiltonian has a long and rich history in the mathematical literature. There even exists a recent review paper devoted to this subject [21]. Let us explain the main points of this history, referring the reader to [21] for more details.

A direct application of the Kato-Rellich theorem yields the essential self-adjointness of the (massive, 3d) Dirac–Coulomb Hamiltonian only for \(|\lambda |<\frac{1}{2}\). This proof is due to Kato [28, 29]. The essential self-adjointness up to the boundary of the “regular region” \(|\lambda |<\frac{\sqrt{3}}{2}\) was proven independently by Gustaffson-Rejtö [26, 34] and Schmincke [36]. They needed to use slightly more refined arguments going beyond to the basic Kato-Rellich theorem. The “distinguished self-adjoint extension” in the region \(\frac{\sqrt{3}}{2}<|\lambda |<1\) was described in several equivalent ways, mostly involving the characterization of the domain, by Schmincke, Wüst, Klaus, Nenciu and others [5, 6, 22, 31, 33, 37, 42, 43]. The characterization of distinguished self-adjoint extensions based on holomorphic families of operators was first proposed by Kato in [30]. Esteban and Loss [18] characterized the distinguished self-adjoint realization at the boundary of the “transitory region”, that is for \(|\lambda |=1\), by using the so-called Hardy-Dirac inequalities. Self-adjoint realizations in the “supercritical region” \(|\lambda |>1\) were first studied by Hogreve in [27], and then (with some corrections) in [22]. The authors of [22] analyze also the second distinguished branch of self-adjoint extensions in the critical region, which they call “mirror distinguished”. [5, 6] include in their analysis a term proportional to \(\frac{1}{r^2}\beta \alpha _ix_i\), which they call “anomalous magnetic”.

Our treatment of Dirac–Coulomb Hamiltonians is quite different from the above references. We use exact solvability to describe rather completely their resolvent, domain and (stationary) scattering theory. We do not add the mass term, which helps with exact solvability and makes possible to use the homogeneity as a good criterion for distinguished realizations. Another concept which we use is that of a holomorphic family of operators, which we view as an important criterion for distinguishing a realization. The mass term is bounded, so it does not affect the basic picture of distinguished realizations. Our analysis includes realizations which are not necessarily self-adjoint, but turn out to be self-transposed with respect to a natural complex bilinear form. Our description of various closed realizations of Dirac–Coulomb Hamiltonians is quite straightforward and involves only elementary functions. We use neither the von Neumann nor the Krein–Vishik theory of self-adjoint extensions, which lead to a rather complicated description of the domains of closed description involving Whittaker functions, see [22, 23].

Our analysis of Dirac–Coulomb Hamiltonians can be viewed as a continuation of a series of papers about holomorphic families of certain one-dimensional Hamiltonians: Bessel operators [3, 12] and Whittaker operators [10, 13].

Let us mention some more papers, where Dirac–Coulomb operators play an important role.

First, there exist a number of papers [7, 16, 24, 39] devoted to the time dependent approach to scattering theory for self-adjoint Dirac Hamiltonians on \(\mathbb {R}^3\) with long range potentials.

There also exists a large and interesting literature devoted to eigenvalues inside a spectral gap of a self-adjoint operator, with massive Dirac–Coulomb Hamiltonians as prime examples [18, 19, 23, 35]. Massless Dirac–Coulomb Hamiltonians do not have a gap, and eigenvalues are possible only in non-self-adjoint nonhomogeneous cases. Nevertheless, we believe that methods of our paper are relevant for the eigenvalue problem in the massive self-adjoint case.

For a study of one-dimensional Dirac operators with locally integrable complex potentials, see [2].

Finally, let us mention another interesting related topic, where the question of distinguished self-adjoint realizations arises: 2-body Dirac–Coulomb Hamiltonians. Their mathematical study was undertaken in [8]. Even though the physical significance of these Hamiltonians is not very clear, they are widely used in quantum chemistry.

Let us briefly describe the organization of our paper. Its main part, that is Sects. 28 describes realizations of 1d Dirac–Coulomb Hamiltonians on \(L^2(\mathbb {R}_+,\mathbb {C}^2)\) focusing on the homogeneous ones. Besides, our paper contains four appendices, which can be read independently.

“Appendix A” first discusses some general concepts related to 1d Dirac operators. Then two special classes of 1d Dirac–Coulomb Hamiltonians are analyzed in detail.

Essentially all papers that we mentioned in our bibliographical sketch treat the three-dimensional case. It was pointed out in [25] that a general d-dimensional spherically symmetric Dirac Hamiltonian can be reduced to a one-dimensional one. We describe this reduction in detail in “Appendix B”. We also analyze its various group-theoretical and differential-geometric aspects, including the relation to Dirac operators on spheres and the famous Lichnerowicz formula. Spectra of the latter are computed in two independent ways and a construction of eigenvectors is presented.

The short “Appendix C” is devoted to the Mellin transformation.

Finally, in “Appendix D” we collect properties of various special functions, mostly, Whittaker functions, which are used in our paper. We mostly follow the conventions of [10, 13].

Remarks About Notation

Symbol \(( \cdot | \cdot )\) is used for standard scalar products on \(L^2\) spaces, linear in the second argument, while \(\langle \cdot | \cdot \rangle \) is used for the analogous bilinear forms in which complex conjugation is omitted:

$$\begin{aligned} \langle f | g \rangle = \int f(x)^{\text {T}} g(x) \text {d}x. \end{aligned}$$
(1.9)

Transpose (denoted by the superscript \(\scriptstyle \text {T}\)) of a densely defined operator is defined in terms of \(\langle \cdot | \cdot \rangle \) in the same way as the adjoint (denoted by \(*\)) is defined in terms of the scalar product. We use superscript \(\text {perp}\) for orthogonal complement with respect to \(\langle \cdot | \cdot \rangle \) and \(\perp \) for orthogonal complement with respect to \(( \cdot | \cdot )\). Overline always denotes complex conjugation; for example, we have \(X^\perp = \overline{X^\text {perp}}\) for a subspace X.

We will write \(\mathbb {R}_+ = ]0, \infty [\), \(\mathbb {C}_{\pm } = \{ z \in \mathbb {C}\, | \, \pm \text {Im}(z) >0 \}\), \(\mathbb {N}=\{0,1,\dots \}\). Omission of zero will be denoted by \(\times \), e.g., \(\mathbb {R}^\times =\mathbb {R}\backslash \{ 0 \}\). Indicator function of a subset \(S \subset \mathbb {R}\) will be denoted by \(\mathbbm {1}_S\). We label elements of the Riemann sphere \(\mathbb {C}\mathbb {P}^1\) using homogeneous coordinates, i.e., \([a:b] \in \mathbb {C}\mathbb {P}^1\) is the complex line spanned by \((a,b) \in \mathbb {C}^2 \backslash \{ 0 \}\).

Operators of multiplication of a function in \(L^2(\mathbb {R}_+, \mathbb {C}^n)\) and \(L^2(\mathbb {R}, \mathbb {C}^n)\) by its argument will be denoted by X and K, respectively. Dilation group action on \(L^2(\mathbb {R}_+,\mathbb {C}^n)\) is defined by \(U_{\tau } f (x) = \text {e}^{\frac{\tau }{2}} f(\text {e}^{\tau } x)\). We denote its self-adjoint generator by A, so that \(U_{\tau } = \text {e}^{\text {i}\tau A}\). Operator O is said to be homogeneous of degree \(\nu \) if \(U_{\tau } O U_{\tau }^{-1} = \text {e}^{\nu \tau } O\). Inversion operator J is defined by \((Jf)(x) = \frac{1}{x} f \left( \frac{1}{x} \right) \). It is unitary and satisfies \(J^2=1\), \(JAJ^{-1}=-A\).

Complex power functions \(z \mapsto z^a\) are holomorphic and defined on \(\mathbb {C}\backslash ] - \infty , 0 ]\). Domains of holomorphy of special functions used in the text are specified in “Appendix D”.

In our paper we will often use the concept of a holomorphic map with values in closed operators, which we now briefly recall [14, 29]. We will give two equivalent definitions of this concept: the first is “more elegant”, the second “more practical”. To formulate the first definition note that the Grassmannian (the set of closed subspaces) \(\mathrm {Grass}(X)\) of a Hilbert space X carries the structure of a complex Banach manifold [17].

Consider Hilbert spaces \(X_2, X_3\) be Hilbert spaces and a complex manifold \(\mathcal {M}\). We say that a function \(\mathcal {M}\ni p\mapsto T_p\) of closed operators \(X_2 \rightarrow X_3\) is holomorphic if and only if \(p \mapsto \mathrm {graph}(T_p) \in \mathrm {Grass}(X_2 \times X_3)\) is a holomorphic map.

Equivalently, \(\mathcal {M}\ni p\mapsto T_p\) is holomorphic if for every \(p_0 \in \mathcal {M}\) there exists a neighborhood \(\mathcal {M}_0\) of \(p_0\) in \(\mathcal {M}\), a Hilbert space \(X_1\) and a holomorphic family \(\mathcal {M}_0\ni p\mapsto S_p\) of bounded injective operators \(S_p : X_1 \rightarrow X_2\) such that \(\text {Ran}(S_p)= \text {Dom}(T_p)\) and \(T_p S_p\) form a holomorphic family of bounded operators.

Blown-Up Quadric

Formal Dirac–Coulomb Hamiltonians depend on parameters \((\omega ,\lambda )\in \mathbb {C}^2\). In order to specify their realizations as closed homogeneous operators, it is necessary to choose a square root of \(\omega ^2-\lambda ^2\). For this reason homogeneous Dirac–Coulomb Hamiltonians are parametrized by points of a certain complex manifold. This section is devoted to its definition and basic properties.

Let us first introduce a certain null quadric in \(\mathbb {C}^3\):

$$\begin{aligned} \mathcal {M}^\text {pre}:= \left\{ (\omega , \lambda , \mu ) \in \mathbb {C}^3 \, | \, \omega ^2 = \lambda ^2 + \mu ^2 \right\} . \end{aligned}$$
(2.1)

By the holomorphic implicit function theorem, \(\mathcal {M}^{\text {pre}}\) is a complex two-dimensional submanifold of \(\mathbb {C}^3\) away from the singular point (0, 0, 0) (also denoted 0 for brevity).

We consider also the so-called blowup of \(\mathcal {M}^\text {pre}\) at the singular point, defined by

$$\begin{aligned} \mathcal {M}= \left\{ (\omega , \lambda , \mu ,[a:b]) \in \mathbb {C}^3 \times \mathbb {C}\mathbb {P}^1 \, | \, \begin{bmatrix} \omega + \lambda &{} \mu \\ \mu &{} \omega - \lambda \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \right\} . \end{aligned}$$
(2.2)

Fibers of the projection map \(\mathcal {M}\rightarrow \mathbb {C}\mathbb {P}^1\) are described by triples \((\omega , \lambda , \mu ) \in \mathbb {C}^3\) subject to two linearly independent linear equations, whose coefficients are holomorphic functions on local coordinate patches of \(\mathbb {C}\mathbb {P}^1\). Therefore, \(\mathcal {M}\) is a holomorphic line bundle over \(\mathbb {C}\mathbb {P}^1\), embedded in the trivial bundle \(\mathbb {C}^3 \times \mathbb {C}\mathbb {P}^1\). In particular it is a two-dimensional complex manifold.

Equation \(\begin{bmatrix} \omega + \lambda &{} \mu \\ \mu &{} \omega - \lambda \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\) has a solution different than \((a,b) = (0,0)\) if and only if the quadratic equation defining \(\mathcal {M}^{\text {pre}}\) is satisfied. Thus, there is a projection map \(\mathcal {M}\rightarrow \mathcal {M}^{\text {pre}}\). Its restriction to the preimage of \(\mathcal {M}^\text {pre}\backslash \{ 0 \}\) is an isomorphism and will be treated as an identification. The preimage of zero, called the zero fiber and denoted \(\mathcal {Z}\), is an isomorphic copy of \(\mathbb {C}\mathbb {P}^1\).

We will often use the short notation \(p=(\omega , \lambda ,\mu ,[a:b])\) for elements of \(\mathcal {M}\). If \(p \notin \mathcal {Z}\), then [a : b] is uniquely determined by \((\omega , \lambda , \mu )\) and we abbreviate \(p = (\omega , \lambda , \mu )\). In turn for p in the zero fiber we write \(p = [a:b]\).

We will now describe useful coordinate systems on \(\mathcal {M}\). The coordinates

$$\begin{aligned} z = \frac{a}{b}, \qquad \omega + \lambda \end{aligned}$$
(2.3)

are valid on \(\{ b \ne 0 \}\) – the open subset of \(\mathcal {M}\) which is the complement of

$$\begin{aligned} \{ b = 0 \} = \{ ( \omega , -\omega , 0,[1:0]) \}_{\omega \in \mathbb {C}}. \end{aligned}$$
(2.4)

More precisely, the following map is an isomorphism of complex manifolds:

$$\begin{aligned} \mathbb {C}^2 \ni (\omega + \lambda , z)\mapsto & {} \left( \frac{(\omega + \lambda ) (1+ z^2)}{2}, \frac{(\omega +\lambda ) (1- z^2)}{2}, - (\omega +\lambda ) z , [z:1] \right) \nonumber \\&\in \{ b \ne 0 \}. \end{aligned}$$
(2.5)

We note that

$$\begin{aligned} z = - \frac{\mu }{\omega + \lambda } = - \frac{\omega - \lambda }{\mu } \end{aligned}$$
(2.6)

whenever the denominators are nonzero.

Analogously, on \(\{ a \ne 0 \}\), the complement of

$$\begin{aligned} \{ a = 0 \} = \{ ( \omega , \omega , 0,[0:1]) \}_{\omega \in \mathbb {C}}, \end{aligned}$$
(2.7)

we use the coordinates \(z^{-1}\) and \(\omega - \lambda \).

Sets \(\{ a \ne 0 \}\), \(\{ b \ne 0 \}\) cover the whole \(\mathcal {M}\). On their intersection we have

$$\begin{aligned} \omega - \lambda = (\omega + \lambda ) z^2. \end{aligned}$$
(2.8)

We note that the locus \(\{ \lambda = 0 \}\) is the union of three Riemann surfaces:

$$\begin{aligned} \{ \lambda = 0 \} = \mathcal {Z}\cup \{ a = b \} \cup \{ a = - b \}. \end{aligned}$$
(2.9)

It is singular at the intersection points:

$$\begin{aligned} \{[1:1] \}=\mathcal {Z}\cap \{ a = b \},\quad \{[1:-1]\}=\mathcal {Z}\cap \{ a = - b \}. \end{aligned}$$
(2.10)

On the other hand, the level sets \(\{ \lambda = \lambda _0 \}\) with \(\lambda _0 \ne 0\) are nonsingular. Similarly, we have

$$\begin{aligned} \{ \mu = 0 \} = \mathcal {Z}\cup \{ a = 0 \} \cup \{ b = 0 \}. \end{aligned}$$
(2.11)

Remark 3

Consider the tautological line bundle \(\mathcal {N}\rightarrow \mathbb {C}\mathbb {P}^1\), i.e., the space of pairs \({\big ((a',b'),[a:b]\big ) \in \mathbb {C}^2 \times \mathbb {C}\mathbb {P}^1}\) such that \((a',b') \in [a:b] \). Setting \(z:=\frac{a'}{b'}\), we obtain two charts \((b',z)\) and \((a',z^{-1})\), which cover \(\mathcal {N}\). The clutching formula for \(\mathcal {N}\) is \(a'=b' z\), which can be compared with the clutching formula (2.8) for \(\mathcal {M}\). Thus, we see that as a holomorphic vector bundle \(\mathcal {M}\) is isomorphic to the tensor square of \(\mathcal {N}\).

Later we will encounter the meromorphic functions on \(\mathcal {M}\)

$$\begin{aligned} N_p^\pm = \frac{z \pm \text {i}}{\Gamma (1 + \mu {\mp } \text {i}\lambda )}. \end{aligned}$$
(2.12)

We define the exceptional sets as their zero loci:

$$\begin{aligned} \mathcal {E}^{\pm } :=&\{ N_p^\pm =0 \} =\bigcup _{n=0}^\infty \mathcal {E}_n^\pm ,\nonumber \\ \mathcal {E}_0^\pm :=&\{p\in \mathcal {M}\,|\, a = {\mp } \text {i}b \}=\{p\in \mathcal {M}\,|\,z={\mp }\text {i}\},\nonumber \\ \mathcal {E}_n^\pm :=&\{ p \in \mathcal {M}\, | \, \mu {\mp } \text {i}\lambda = -n \},\quad n=1,2,\dots . \end{aligned}$$
(2.13)

Away from \(\mathcal {Z}\), the condition \(p\in \mathcal {E}_0^\pm \) is equivalent to \(\mu {\mp } \text {i}\lambda =0\). Thus, for \(p \notin \mathcal {Z}\) we have \(p \in \mathcal {E}^{\pm }\) if and only if \(\mu {\mp } \text {i}\lambda \in - {\mathbb {N}}\). Moreover,

$$\begin{aligned} \mathcal {E}^{\pm }\cap \mathcal {Z}=\mathcal {E}_0^\pm \cap \mathcal {Z}= \{ [{\mp } \text {i}: 1] \}.\end{aligned}$$
(2.14)

In particular \(\mathcal {Z}\cap \mathcal {E}^+\cap \mathcal {E}^- = \emptyset \). Clearly, the sets \(\mathcal {E}_n^\pm \), \(n=0,1,2,\dots \), are connected components of \(\mathcal {E}^\pm \). Each \(\mathcal {E}_n^\pm \) is isomorphic to \(\mathbb {C}\). Indeed, \(\mathcal {E}_0^\pm \) is a fiber of \(\mathcal {M}\rightarrow \mathbb {C}\mathbb {P}^1\) and \(\mathcal {E}_n^\pm \) with \(n \ge 1\) is globally parametrized by \(\omega \).

Lemma 4

\(\mathcal {E}^+ \cap \mathcal {E}^-\) is a countably infinite discrete subset of \(\mathcal {M}\) on which \(2 \mu + 1 \in - {\mathbb {N}}\). In particular \(\mu \le - \frac{1}{2}\).

Proof

Suppose that \(p \in \mathcal {M}\) is such that \(\mu + \text {i}\lambda = -n\), \(\mu - \text {i}\lambda = -m\) with \(n,m \in {\mathbb {N}}\). Then

$$\begin{aligned} (\omega , \lambda , \mu ) = \left( \pm nm, \frac{m-n}{2 \text {i}} , - \frac{m+n}{2} \right) , \qquad (n,m) \in \mathbb {N}^2 \backslash \{ (0,0) \}, \end{aligned}$$
(2.15)

from which the discreteness and countability of \(\mathcal {E}^+ \cap \mathcal {E}^-\) is clear. If both nm are zero, then \(\mu = \lambda =0\) and hence also \(\omega = 0\). In this case we have \(p \in \mathcal {Z}\cap \mathcal {E}^+ \cap \mathcal {E}^- = \emptyset \)—contradiction. Thus, at least one of nm is nonzero, and we have \(2 \mu + 1 = 1 - n - m \in - {\mathbb {N}}\). Conversely, if \((n,m) \in {\mathbb {N}}^2\) is different than (0, 0), then (2.15) defines one or two (if \(nm \ne 0\)) points of \(\mathcal {E}^+ \cap \mathcal {E}^-\), so this set is infinite. \(\square \)

We define the principal scattering amplitude as the ratio

$$\begin{aligned} S_p = \frac{N_p^-}{N_p^+} = \frac{z-\text {i}}{z+\text {i}} \frac{\Gamma (1 + \mu - \text {i}\lambda )}{\Gamma (1 + \mu + \text {i}\lambda )} = \frac{(\omega - \lambda + \text {i}\mu ) \Gamma (1 + \mu - \text {i}\lambda )}{(\omega - \lambda - \text {i}\mu ) \Gamma (1 + \mu + \text {i}\lambda )}. \end{aligned}$$
(2.16)

It satisfies \(\overline{S_{{\overline{p}}}} = S_p^{-1}\); hence, it has a unit modulus for \(p = {\overline{p}}\). Furthermore,

$$\begin{aligned}&\mathcal {E}^- \backslash \mathcal {E}^+ = \{ S_p = 0 \}, \qquad \mathcal {E}^+ \backslash \mathcal {E}^- = \{ S_p = \infty \}, \nonumber \\&\quad \mathcal {E}^- \cap \mathcal {E}^+ = \{ S_p \text { indeterminate} \}. \end{aligned}$$
(2.17)

We introduce an involution on \(\mathcal {M}\) by

$$\begin{aligned} \tau (\omega , \lambda , \mu , [a:b]) = (\omega , \lambda , - \mu , [-a:b]). \end{aligned}$$
(2.18)

Eigenfunctions and Green’s Kernels

Zero Energy

The 1d Dirac–Coulomb Hamiltonian with parameters \(\omega ,\lambda \in \mathbb {C}\) is given by the expression

$$\begin{aligned} D_{\omega , \lambda } = \begin{bmatrix} -\frac{\lambda +\omega }{x} &{} - \partial _x \\ \partial _x &{} -\frac{\lambda -\omega }{x} \end{bmatrix}. \end{aligned}$$
(3.1)

When we consider (3.1) as acting on distributions on \(\mathbb {R}_+\), we will call it the formal operator. In what follows we will define various realizations of this operator, with domain and range contained in \(L^2(\mathbb {R}_+,\mathbb {C}^2)\), preferably closed. They will have additional indices.

First consider its eigenequation for eigenvalue zero

$$\begin{aligned} D_{\omega ,\lambda }\xi =0. \end{aligned}$$
(3.2)

The space of distributions on \(\mathbb {R}_+\) solving (3.2) will be denoted \(\text {Ker}(D_{\omega ,\lambda })\). The following lemma shows that \(\text {Ker}(D_{\omega ,\lambda })\) consists of smooth solutions.

Lemma 5

Let f be a distributional solution on \(\mathbb {R}_+\) of the equation \(f'(x) = M(x)f(x)\) for some \(M \in C^{\infty }(\mathbb {R}_+, \mathrm {End}(\mathbb {C}^n))\). Then f is a smooth function.

Proof

Fix \(x_0 \in \mathbb {R}_+\) and \(\epsilon \in \left] 0 , \frac{x_0}{2} \right[\). We choose \(\chi _2 \in C_c^{\infty }(\mathbb {R}_+)\) equal to 1 on \([x_0 - 2 \epsilon , x_0 + 2\epsilon ]\) and \(\chi _1 \in C_c^{\infty }(\mathbb {R}_+)\) supported in \([x_0 - 2 \epsilon , x_0 + 2\epsilon ]\) and equal to 1 on \([x_0 - \epsilon , x_0 + \epsilon ]\). Clearly \(\chi _2 \chi _1 = \chi _1\) and \(\chi _2 \chi _1' = \chi _1'\). Put \(f_j = \chi _j f\) for \(j=1,2\). Since \(f_2\) is compactly supported, it belongs to \(H^s(\mathbb {R}_+,\mathbb {C}^n)\) for some \(s \in \mathbb {R}\). We have \(f_1 = \chi _1 f_2\), so also \(f_1 \in H^s(\mathbb {R}_+,\mathbb {C}^n)\). Now evaluate

$$\begin{aligned} f_1 ' = \chi _1' f + \chi _1 f' = \chi _2 \chi _1' f + \chi _2 \chi _1 M f = (\chi _1' + \chi _1 M) f_2. \end{aligned}$$
(3.3)

Since \(\chi _1' + \chi _1 M \in C_c^{\infty }(\mathbb {R}_+)\), this implies that \(f_1 \in H^{s+1}(\mathbb {R}_+,\mathbb {C}^n)\). Next we may repeat this argument with \(\frac{\epsilon }{2}\) playing the role of new \(\epsilon \), \(\chi _1\) as the new \(\chi _2\) and arbitrarily chosen new \(\chi _1\). Then the new \(f_1\) is in \(H^{s+2}(\mathbb {R}_+, \mathbb {C}^n)\). Proceeding like this inductively we conclude that for every \(s \in \mathbb {R}\) there exists \(\chi \in C_c^{\infty }(\mathbb {R}_+)\) equal to 1 on a neighborhood of \(x_0\) such that \(\chi \, f\) belongs to \(H^s({\mathbb {R}}_+,\mathbb {C}^n)\). Taking \(s> \frac{1}{2}+k\) we conclude from Sobolev embeddings that f is of class \(C^k\) on a neighborhood of \(x_0\), perhaps after modifying it on a set of measure zero. Since this is true for every \(k \in {\mathbb {N}}\) and every \(x_0 \in \mathbb {R}_+\), f is smooth. \(\square \)

For \(p\in \mathcal {M}_{} \), we introduce two types of solutions of (3.2):

$$\begin{aligned} \eta _p^\uparrow (x)&:= \frac{x^{\mu }}{\omega + \lambda } \begin{bmatrix}-\mu \\ \omega +\lambda \end{bmatrix}= x^{\mu }\begin{bmatrix}z \\ 1\end{bmatrix}, \end{aligned}$$
(3.4a)
$$\begin{aligned} \eta _p^\downarrow (x)&:= \frac{x^{\mu }}{\omega - \lambda }\begin{bmatrix}\omega -\lambda \\ -\mu \end{bmatrix}= x^{\mu }\begin{bmatrix}1 \\ z^{-1} \end{bmatrix}. \end{aligned}$$
(3.4b)

They are nowhere vanishing meromorphic functions of p for every x:

$$\begin{aligned} \mathcal {M}\ni p \mapsto \eta _p^\uparrow (x)&\quad \text {has a pole on } \{ b = 0 \}, \\ {\mathcal {M}}_{} \ni p\mapsto \eta _p^\downarrow (x)&\quad \text {has a pole on } \{ a = 0 \}. \end{aligned}$$

On \(\{ a \ne 0 \} \cap \{ b \ne 0 \}\) we have \(\eta _p^{\downarrow } = z^{-1} \eta _p^{\uparrow }\).

There exist also exceptional solutions, defined only for \(\mu =0\):

$$\begin{aligned} \vartheta _{\omega }^\uparrow (x)&:= -\ln (x) \begin{bmatrix}0\\ 2\omega \end{bmatrix} +\begin{bmatrix}1\\ 0\end{bmatrix},\quad \omega -\lambda =0; \end{aligned}$$
(3.5a)
$$\begin{aligned} \vartheta _{ \omega }^\downarrow (x)&:= -\ln (x) \begin{bmatrix}2\omega \\ 0\end{bmatrix} +\begin{bmatrix}0\\ 1\end{bmatrix},\quad \omega +\lambda =0 .\end{aligned}$$
(3.5b)

The nullspace of \(D_{\omega ,\lambda }\), that is, \(\text {Ker}(D_{\omega ,\lambda })\) has the following bases:

$$\begin{aligned} \mu \ne 0 :&\qquad \big ( \eta _{\omega ,\lambda ,\mu }^\uparrow ,\quad \eta _{\omega ,\lambda ,-\mu }^\uparrow \big )\quad \text {and}\quad \big ( \eta _{\omega ,\lambda ,\mu }^\downarrow ,\quad \eta _{\omega ,\lambda ,-\mu }^\downarrow \big ),\\ \omega =\lambda \ne 0 :&\qquad \big ( \eta _{\omega ,\omega ,0}^\uparrow ,\quad \vartheta _{\omega }^\uparrow \big ),\\ \omega =-\lambda \ne 0 :&\qquad \big ( \eta _{\omega ,- \omega ,0}^\downarrow ,\quad \vartheta _{\omega }^\downarrow \big ),\\ ( \omega ,\lambda )=(0,0):&\qquad \big ( \vartheta _{0}^\uparrow ,\quad \vartheta _{0}^\downarrow \big ) = \left( \begin{bmatrix}1\\ 0\end{bmatrix},\quad \begin{bmatrix}0\\ 1\end{bmatrix}\right) . \end{aligned}$$

The canonical bisolution of \(D_{\omega ,\lambda }\) (A.5) at \(k=0\) takes the form

$$\begin{aligned} G_{\omega ,\lambda }^\leftrightarrow (0;x,y)&=\frac{1}{2}\begin{bmatrix} \frac{\omega -\lambda }{\mu } \Big (\Big (\frac{x}{y}\Big )^\mu -\Big (\frac{y}{x}\Big )^\mu \Big )&{} \Big (\frac{x}{y}\Big )^\mu +\Big (\frac{y}{x}\Big )^\mu \\ -\Big (\frac{x}{y}\Big )^\mu -\Big (\frac{y}{x}\Big )^\mu &{}-\frac{\omega +\lambda }{\mu } \Big (\Big (\frac{x}{y}\Big )^\mu -\Big (\frac{y}{x}\Big )^\mu \Big ) \end{bmatrix}. \end{aligned}$$
(3.6)

Nonzero Energy

Now consider the eigenequation for the eigenvalue \(k \in \mathbb {C}^\times \):

$$\begin{aligned} ( D_{\omega , \lambda } -k)f=0. \end{aligned}$$
(3.7)

Acting on (3.7) with \( D_{\omega , - \lambda }+k\) we obtain

$$\begin{aligned} \begin{bmatrix}- \partial _x^2 + \frac{\omega ^2 - \lambda ^2}{x^2} - \frac{2 \lambda k}{x} - k^2 &{}\frac{\omega - \lambda }{x^2}\\ \frac{\omega + \lambda }{x^2}&{}- \partial _x^2 + \frac{\omega ^2 - \lambda ^2}{x^2} - \frac{2 \lambda k}{x} - k^2 \end{bmatrix}f(x)=0. \end{aligned}$$
(3.8)

At first we focus on the case \(\mu ^2=\omega ^2 - \lambda ^2 \ne 0\), in which \(\begin{bmatrix}0 &{}\omega - \lambda \\ \omega + \lambda &{}0 \end{bmatrix}\) is a diagonalizable matrix. Decomposing f(x) in its eigenbasis

$$\begin{aligned} f(x) = f_+(x) \begin{bmatrix} \omega - \lambda \\ \mu \end{bmatrix} + f_-(x) \begin{bmatrix} \omega - \lambda \\ -\mu \end{bmatrix} \end{aligned}$$
(3.9)

we find that functions \(f_{\pm } (x)\) satisfy the Whittaker equations

$$\begin{aligned} \left( - \partial _x^2 + \frac{\left( \mu \pm \frac{1}{2} \right) ^2 - \frac{1}{4}}{x^2} - \frac{2 \lambda k}{x} - k^2 \right) f_{\pm }(x) =0. \end{aligned}$$
(3.10)

This second-order differential equation is satisfied by the Whittaker functions (D.15) and (D.18):

$$\begin{aligned} f_{\pm }(x) = c_{\pm , 1} \mathcal {I}_{- \text {i}\lambda , \mu \pm \frac{1}{2}}(2 \text {i}kx) + c_{\pm , 2} \mathcal {K}_{- \text {i}\lambda , \mu \pm \frac{1}{2}}(2 \text {i}kx). \end{aligned}$$
(3.11)

For generic values of parameters, the four functions appearing in (3.11) are linearly independent and thus (3.11) is the general solution of (3.8). Inspection of its expansion for \(x \rightarrow 0\) reveals that (again, for generic parameters) it is annihilated by \( D_{\omega , \lambda } -k\) if and only if

$$\begin{aligned} \text {i}\omega \, c_{-,1} = c_{+,1}, \qquad \omega \, c_{-,2} = ( \lambda + \text {i}\mu ) c_{+,2}. \end{aligned}$$
(3.12)

Remark 6

Equation (3.7) simplifies for \(\mu =0\), but instead of treating it separately we will construct solutions valid on the whole \(\mathcal {M}\) by analytic continuation. For similar reasons we disregard non-generic cases mentioned above Equation (3.12).

Let us introduce a family of solutions of the eigenequation (3.7) defined for \(k \in \mathbb {C}\backslash [0, \text {i}\infty [\):

$$\begin{aligned} \xi _p^-(k,x)= & {} \frac{\Gamma (1 + \mu + \text {i}\lambda )}{2 \mu (\omega - \lambda + \text {i}\mu )} \left( \text {i}\omega \mathcal {I}_{- \text {i}\lambda , \mu + \frac{1}{2}} (2 \text {i}k x) \begin{bmatrix} \omega - \lambda \\ \mu \end{bmatrix} \right. \nonumber \\&\quad +\, \left. \mathcal {I}_{- \text {i}\lambda , \mu - \frac{1}{2}} (2 \text {i}k x) \begin{bmatrix} \omega - \lambda \\ - \mu \end{bmatrix} \right) , \end{aligned}$$
(3.13a)
$$\begin{aligned} \zeta _{p}^-(k,x)= & {} \frac{\omega \, \mathcal {K}_{- \text {i}\lambda , \mu + \frac{1}{2}}(2 \text {i}k x)}{\mu (\omega - \lambda - \text {i}\mu )} \begin{bmatrix} \omega - \lambda \\ \mu \end{bmatrix} \nonumber \\&\quad +\, \frac{(\lambda + \text {i}\mu ) \mathcal {K}_{- \text {i}\lambda , \mu - \frac{1}{2}}(2 \text {i}k x)}{\mu (\omega - \lambda - \text {i}\mu )} \begin{bmatrix} \omega - \lambda \\ - \mu \end{bmatrix}. \end{aligned}$$
(3.13b)

As an alternative to the presented derivation, one may check directly that they satisfy (3.7) using recursion relations from “Appendix D.4”.

The second family, defined for \(k \in \mathbb {C}\backslash [0, - \text {i}\infty [\), is obtained by reflection:

$$\begin{aligned} \xi _p^+(k,x) = \overline{\xi _{\overline{p}}^-({\overline{k}} , x)}, \qquad \zeta _p^+(k,x) = \overline{\zeta _{\overline{p}}^-({\overline{k}} , x)}. \end{aligned}$$
(3.14)

Explicit expressions in terms of Whittaker functions take the form

$$\begin{aligned} \xi _p^+(k,x)= & {} \frac{\Gamma (1 + \mu - \text {i}\lambda )}{2 \mu (\omega - \lambda - \text {i}\mu )} \nonumber \\&\quad \times \, \left( -\text {i}\omega \mathcal {I}_{ \text {i}\lambda , \mu + \frac{1}{2}} (-2 \text {i}k x) \begin{bmatrix} \omega - \lambda \\ \mu \end{bmatrix} + \mathcal {I}_{ \text {i}\lambda , \mu - \frac{1}{2}} (-2 \text {i}k x) \begin{bmatrix} \omega - \lambda \\ - \mu \end{bmatrix} \right) ,\nonumber \\ \end{aligned}$$
(3.15a)
$$\begin{aligned} \zeta _{p}^+(k,x)= & {} \frac{\omega \, \mathcal {K}_{ \text {i}\lambda , \mu + \frac{1}{2}}(-2 \text {i}k x)}{\mu (\omega - \lambda + \text {i}\mu )} \begin{bmatrix} \omega - \lambda \\ \mu \end{bmatrix} \nonumber \\&\quad +\, \frac{(\lambda - \text {i}\mu ) \mathcal {K}_{ \text {i}\lambda , \mu - \frac{1}{2}}(-2 \text {i}k x)}{\mu (\omega - \lambda + \text {i}\mu )} \begin{bmatrix} \omega - \lambda \\ - \mu \end{bmatrix}. \end{aligned}$$
(3.15b)

Lemma 7

Let us fix kx. \(\xi ^{+}_p(k,x)\) and \(\xi ^{-}_p(k,x)\) are meromorphic functions of \(p \in \mathcal {M}\), nonsingular away from \(\mathcal {E}^+\) and \(\mathcal {E}^-\), respectively. \(\zeta _p^+(k,x)\) and \(\zeta _p^-(k,x)\) are holomorphic functions on the whole \(\mathcal {M}\). Furthermore, \(\zeta _p^\pm (\cdot )\) satisfy \(\zeta _p^\pm = \zeta _{\tau (p)}^\pm \), where \(\tau \) was defined in (2.18), and are nonzero functions for every \(p \in \mathcal {M}\).

Proof

It is sufficient to prove the claim for the family with superscript minus. Meromorphic dependence on p is clear. Definitions of \(\xi _p^-\) and \(\zeta _p^-\) can be manipulated to the form

$$\begin{aligned} \xi _{p}^-(k,x)= & {} \text {i}\, \mathcal {I}_{- \text {i}\lambda , \mu + \frac{1}{2}}(2 \text {i}k x) \frac{1}{ N_p^-} \begin{bmatrix} -1 \\ z \end{bmatrix} \nonumber \\&\quad +\, \frac{\mathcal {I}_{- \text {i}\lambda , \mu - \frac{1}{2}}(2 \text {i}k x) - \text {i}\lambda \, \mathcal {I}_{- \text {i}\lambda ,\mu + \frac{1}{2}}(2 \text {i}k x)}{\mu } \frac{1}{ N_p^-}\begin{bmatrix} z \\ 1 \end{bmatrix}, \end{aligned}$$
(3.16a)
$$\begin{aligned} \zeta _{p}^-(k,x)= & {} \mathcal {K}_{- \text {i}\lambda , \mu + \frac{1}{2}} (2 \text {i}k x) \begin{bmatrix} \text {i}\\ 1 \end{bmatrix} \nonumber \\&\quad -\, \frac{(\omega +\lambda ) (z + \text {i})}{2} \frac{\mathcal {K}_{- \text {i}\lambda , \mu - \frac{1}{2}}(2 \text {i}k x) - \mathcal {K}_{- \text {i}\lambda , \mu + \frac{1}{2}}(2 \text {i}k x) }{\mu } \begin{bmatrix} z \\ 1 \end{bmatrix} \end{aligned}$$
(3.16b)
$$\begin{aligned}&= \mathcal {K}_{- \text {i}\lambda , \mu -\frac{1}{2}} (2 \text {i}k x) \begin{bmatrix} \text {i}\\ 1 \end{bmatrix} \nonumber \\&\quad -\, \frac{(\omega +\lambda ) (-z + \text {i})\big (\mathcal {K}_{- \text {i}\lambda , \mu - \frac{1}{2}}(2 \text {i}k x) - \mathcal {K}_{- \text {i}\lambda , \mu + \frac{1}{2}}(2 \text {i}k x)\big ) }{2\mu } \begin{bmatrix} -z \\ 1 \end{bmatrix}.\nonumber \\ \end{aligned}$$
(3.16c)

Functions \(\mu \mapsto \frac{\mathcal {I}_{- \text {i}\lambda , \mu - \frac{1}{2}}(2 \text {i}k x) - \text {i}\lambda \, \mathcal {I}_{- \text {i}\lambda ,\mu + \frac{1}{2}}(2 \text {i}k x)}{\mu }\) and \(\mu \mapsto \frac{\mathcal {K}_{- \text {i}\lambda , \mu - \frac{1}{2}}(2 \text {i}k x) - \mathcal {K}_{- \text {i}\lambda , \mu + \frac{1}{2}}(2 \text {i}k x) }{\mu }\) have removable singularities at \(\mu = 0\), as seen from identities (D.17), (D.19a). Therefore, \(\zeta _p^-(k,x)\) is regular for \(z \ne \infty \). If in addition \(p \notin \mathcal {E}^-\), then also \((N_p^-)^{-1}\) and hence \(\xi _p^-\) is nonsingular.

Next write \((\omega +\lambda ) (z+ \text {i}) \begin{bmatrix} z \\ 1 \end{bmatrix} = (\omega - \lambda ) (1+ \text {i}z^{-1}) \begin{bmatrix} 1 \\ z^{-1} \end{bmatrix}\) in (3.16a). Then it is clear that \(\zeta _p^-(k,x)\) is nonsingular for \(z=\infty \).

Moreover, \(\frac{z}{ N_p^-} = \Gamma (1 + \mu + \text {i}\lambda )(1- \text {i}z^{-1})^{-1}\) is nonsingular for \(z=\infty \) if \(p\not \in \mathcal {E}_0^-\). Hence, \(\xi _{p}^-(k,x) \) is nonsingular for \(z=\infty \) if \(p \notin \mathcal {E}^-\).

The statement about the symmetry \(\mu \rightarrow -\mu \) follows from the comparison of (3.16a) and (3.16c). The last claim follows from (3.19) below. \(\square \)

Remark 8

Proof of Lemma 7 shows that functions \(\frac{\xi _p^{\pm }(k,x)}{\Gamma (1+ \mu {\mp } \text {i}\lambda )}\) are singular on subsets of \(\mathcal {M}\) smaller than \(\mathcal {E}^{\pm }\), namely on \(\mathcal {E}_0^\pm \). Moreover, \(\frac{(\omega - \lambda {\mp } \text {i}\mu ) \xi _p^{\pm }(k,x)}{\Gamma (1+ \mu {\mp } \text {i}\lambda )}\) is holomorphic everywhere on \(\mathcal {M}\); however, it vanishes on \(\mathcal {Z}\cup \{ a = 0 \}\).

Near the origin, \(\xi _p^\pm \) has the leading term proportional to \((kx)^\mu \), except for \(2 \mu +1 \in - {\mathbb {N}}\):

$$\begin{aligned} \xi _p^\pm (k,x) \sim \frac{1}{N_p^{\pm }} \frac{({\mp } 2 \text {i}k x)^{\mu }}{\Gamma (2 \mu +1)} \begin{bmatrix} z \\ 1 \end{bmatrix}+ O(( k x)^{\mu +1}) \end{aligned}$$
(3.17)

If \(k \in \mathbb {C}_\pm \), it grows exponentially at infinity:

$$\begin{aligned} \xi _p^\pm (k,x) \sim \frac{1}{2} \text {e}^{{\mp } \text {i}k x} ({\mp } 2 \text {i}k x)^{{\mp } \text {i}\lambda } \begin{bmatrix} 1 \\ {\mp } \text {i}\end{bmatrix}+ O(\text {e}^{{\mp } \text {i}k x}( kx)^{{\mp } \text {i}\lambda -1}). \end{aligned}$$
(3.18)

Under the same assumption, \(\zeta _p^\pm \) is exponentially decaying:

$$\begin{aligned} \zeta _p^\pm (k,x) \sim \text {e}^{\pm \text {i}k x} ({\mp } 2 \text {i}k x)^{\pm \text {i}\lambda } \begin{bmatrix} {\mp } \text {i}\\ 1 \end{bmatrix}+O(\text {e}^{\pm \text {i}k x} ( kx)^{\pm \text {i}\lambda -1}). \end{aligned}$$
(3.19)

Behavior of this function near the origin is much more complicated, see (D.24). Here we note only that for \(\text {Re}(\mu ) >0\) one has

$$\begin{aligned} \zeta _{p}^\pm (k,x) \sim \frac{N_p^{\pm }}{2} \Gamma (2 \mu + 1) ({\mp } 2 \text {i}k x )^{- \mu } \begin{bmatrix} -1 \\ z^{-1} \end{bmatrix} + o (( kx)^{- \mu }). \end{aligned}$$
(3.20)

For \(k \in \mathbb {C}\backslash \text {i}\mathbb {R}\) both families of solutions are defined. The following lemma provides relations between them. It is convenient to introduce \(\varepsilon _k = \text {sgn}(\text {Re}(k))\), which distinguishes connected components of \(\mathbb {C}\backslash \text {i}\mathbb {R}\).

Lemma 9

For every \(k \in \mathbb {C}\backslash \text {i}\mathbb {R}\) we have

$$\begin{aligned} \xi _p^+(k,x)&= \text {e}^{- \text {i}\varepsilon _k \pi \mu } S_p \, \xi _p^- (k,x), \end{aligned}$$
(3.21a)
$$\begin{aligned} \xi _p^+(k,x)&=\frac{\text {e}^{-\varepsilon _k \pi \lambda }}{2\text {i}}\big (\zeta _p^-(k,x)-\text {e}^{-\text {i}\varepsilon _k \pi \mu }S_p\zeta _p^+(k,x)\big ) , \end{aligned}$$
(3.21b)
$$\begin{aligned} \xi _p^-(k,x)&=\frac{\text {e}^{-\varepsilon _k \pi \lambda }}{2\text {i}}\big (\text {e}^{\text {i}\varepsilon _k \pi \mu }S_p^{-1}\zeta _p^-(k,x)-\zeta _p^+(k,x)\big ), \end{aligned}$$
(3.21c)
$$\begin{aligned} \zeta _p^\pm (k,x)&= {\mp } 2 \text {i}\text {e}^{\varepsilon _k \pi \lambda } \xi _p^{\mp }(k,x) + \text {e}^{\pm \text {i}\varepsilon _k \pi \mu } (S_p)^{{\mp } 1} \zeta _p^{\mp }(k,x). \end{aligned}$$
(3.21d)

Proof

Equation (3.21a) follows immediately from (D.16a). To derive (3.21b), we express \(\xi _p^\pm \) and \(\zeta _p^\pm \) in terms of trigonometric Whittaker functions and use the connection formula (D.33). Then (3.21c) is obtained by reflection or by combining with (3.21a). Equation (3.21d) is obtained from (3.21b) and (3.21c) by inverting and multiplying \(2 \times 2\) matrices. \(\square \)

Lemma 10

\(\xi _p^{\pm }\) and \(\xi _{\tau (p)}^\pm \), two eigenvectors of the monodromy, can be used to express \(\zeta _p^{\pm }\):

$$\begin{aligned} \zeta _p^\pm (k,x) = - \frac{2 \pi \omega }{\Gamma (1 + \mu {\mp } \text {i}\lambda ) \Gamma (1 - \mu {\mp } \text {i}\lambda )} \frac{\xi _p^\pm (k,x) - \xi _{\tau (p)}^\pm (k,x)}{\sin (2 \pi \mu )}. \end{aligned}$$
(3.22)

The analytic continuation of \(\zeta _p^\pm \) along a loop winding around the origin counterclockwise gives

$$\begin{aligned} \zeta _p^\pm (\text {e}^{2 \pi \text {i}} k , x) = \text {e}^{-2 \pi \text {i}\mu } \zeta _p^\pm (k,x) - \frac{4 \pi \text {i}\omega }{\Gamma (1 + \mu {\mp } \text {i}\lambda ) \Gamma (1 - \mu {\mp } \text {i}\lambda )} \xi _p^\pm (k,x). \end{aligned}$$
(3.23)

Proof

Relation (3.22) may be derived from (D.18). Then (3.23) follows immediately. \(\square \)

Lemma 11

The following relations hold:

$$\begin{aligned} \det \begin{bmatrix} \xi _{p}^\pm (k,x)&\zeta _{p}^\pm (k,x) \end{bmatrix}&=1, \end{aligned}$$
(3.24a)
$$\begin{aligned} \det \begin{bmatrix} \zeta _{p}^+(k,x)&\zeta _{p}^-(k,x) \end{bmatrix}&= -2\text {i}\text {e}^{\varepsilon _k\pi \lambda }. \end{aligned}$$
(3.24b)

In particular, \(\xi _p^\pm (k, \cdot ), \zeta _p^\pm (k, \cdot )\) form a basis of solutions of (3.7) for \(p \notin \mathcal {E}^\pm \) and \(k \notin [0, {\mp } \text {i}\infty [\), while \(\zeta _p^+(k, \cdot )\) and \(\zeta _p^-(k, \cdot )\) form a basis whenever \(k \notin \text {i}\mathbb {R}\).

Proof

Equation (3.7) may be rewritten in the form \(f'(x) = M(x) f(x)\), where M(x) is a traceless matrix. Therefore, for any two solutions fg the determinant \(\det \begin{bmatrix} f(x)&g(x) \end{bmatrix}\) is independent of x. To calculate it for \(f = \xi _{p}^\pm (k,\cdot )\), \(g = \zeta _{p}^\pm (k, \cdot )\), we use their asymptotic forms for \(x \rightarrow 0\). By holomorphy, it is sufficient to carry out the computation for \(\text {Re}(\mu ) > 0\). Then we may use (3.17) and (3.20). To obtain (3.24b), we combine (3.21b) with (3.24a). \(\square \)

We remark that restrictions on k in Lemma 11 may be omitted if the functions \(\xi _p^\pm \) and \(\zeta _p^\pm \) are analytically continued in suitable way.

Lemma 12

The following relation holds for \(p \in \mathcal {E}^{\pm }\):

$$\begin{aligned} \zeta ^{\pm }_p(k,x) = {\mp } 2 \text {i}\text {e}^{{\mp } \text {i}\pi (\mu {\mp } \text {i}\lambda )} (S_p)^{{\mp } 1} \xi ^{\pm }_p(k,x). \end{aligned}$$
(3.25)

In particular \((S_p)^{{\mp } 1} \xi ^{\pm }_p(k,x)\) is nonsingular on \(\mathcal {E}^{\pm }\).

Proof

It is sufficient to consider the lower sign. If \(p \in \mathcal {E}^-\), then either \(1 + \mu + \text {i}\lambda \in - {\mathbb {N}}\) or \(z=\text {i}\) (and hence \(\mu + \text {i}\lambda =0\)). In the former case we use (D.21) for both terms in (3.13b). In the latter case (D.21) may be used only for the second term, but the first term in both (3.13a) and (3.13b) vanishes. This establishes (3.25). \(\square \)

The following function will be called the two-sided Green’s kernel. It is defined if \(k \in \mathbb {C}_\pm \) and \(p \notin \mathcal {E}^{\pm }\):

$$\begin{aligned} G_p^{\bowtie }(k;x,y)= & {} -\mathbbm {1}_{\mathbb {R}_+}(y -x) \xi _{p}^\pm (k,x) \zeta _{p}^\pm (k,y)^\text {T}\nonumber \\&\quad -\, \mathbbm {1}_{\mathbb {R}_+}(x-y ) \zeta _{p}^\pm (k,x) \xi _{p}^\pm (k,y)^\text {T}. \end{aligned}$$
(3.26)

It is a holomorphic function of \(p \in \mathcal {M}\backslash \mathcal {E}^{\pm }\) satisfying

$$\begin{aligned} G_{p}^{\bowtie }(k;x,y ) = G_{p}^{\bowtie }(k;y ,x)^\text {T}\qquad \text {and} \qquad \overline{G_{p}^{\bowtie }(k,x,y )} = G_{\overline{p}}^{\bowtie }(\overline{k};x,y ). \end{aligned}$$
(3.27)

Later on, with some restrictions on parameters, it will be interpreted as the resolvent of certain closed realizations of \(D_p\).

Minimal and Maximal Operators

We consider the operator

$$\begin{aligned} D_{\omega , \lambda } = \begin{bmatrix} -\frac{\lambda +\omega }{x} &{} - \partial _x \\ \partial _x &{} -\frac{\lambda -\omega }{x} \end{bmatrix}. \end{aligned}$$
(4.1)

on distributions on \(\mathbb {R}_{+} = ]0,\infty [\) valued in \(\mathbb {C}^2\). We will construct out of it several densely defined operators on \(L^2(\mathbb {R}_+,\mathbb {C}^2)\).

Firstly, we let \(D_{\omega , \lambda }^\mathrm {pre}\) be the restriction of \( D_{\omega , \lambda }\) to \(C_c^{\infty }=C_c^{\infty }(\mathbb {R}_+, \mathbb {C}^2)\), called the preminimal realization of \(D_{\omega ,\lambda }\). We have \(D_{ \overline{\omega }, {\overline{\lambda }}}^\mathrm {pre} \subset D_{\omega , \lambda }^{\mathrm {pre}*}\), so \(D_{\omega , \lambda }^{\mathrm {pre}*}\) is densely defined. Thus, \(D_{\omega , \lambda }^\mathrm {pre}\) is closable. Its closure will be denoted by \(D_{\omega , \lambda }^{\min }\). Next, \(D_{\omega , \lambda }^{\max }\) is defined as the restriction of \( D_{\omega , \lambda }\) to \(\text {Dom}(D_{\omega , \lambda }^{\max }) = \{ f \in L^2(\mathbb {R}_+,\mathbb {C}^2) \, | \, D_{\omega , \lambda } f \in L^2(\mathbb {R}_+,\mathbb {C}^2) \}\). It is easy to check that \(D_{\omega , \lambda }^{\min } \subset D_{\omega ,\lambda }^{\max } = D_{ \overline{\omega }, \overline{\lambda }}^{\mathrm {pre}*}\). Furthermore, \(\overline{D_{\omega , \lambda }^\mathrm {pre}} = D_{ \overline{\omega }, {\overline{\lambda }}}^\mathrm {pre}\) and analogously for \(D_{\omega , \lambda }^{\min }\) and \(D_{\omega , \lambda }^{\max }\). As a consequence,

$$\begin{aligned} D_{\omega ,\lambda }^{\min \text {T}}=D_{\omega ,\lambda }^{\max },\qquad D_{\omega ,\lambda }^{\max \text {T}}=D_{\omega ,\lambda }^{\min }. \end{aligned}$$
(4.2)

Operators \(D_{\omega , \lambda }^\text {pre}, D_{\omega , \lambda }^{\min }\) and \(D_{\omega , \lambda }^{\max }\) are all homogeneous of order \(-1\).

We choose \(\mu \in \mathbb {C}\) satisfying \(\mu ^2 = \omega ^2 - \lambda ^2\). Note that in general \(\mu \) is not uniquely determined by \(\omega , \lambda \). For the moment it does not matter which one we take.

Theorem 13

  1. 1.

    If \(|\text {Re}(\mu )|\ge \frac{1}{2}\), then

    $$\begin{aligned} \text {Dom}(D_{\omega , \lambda }^{\max })=\text {Dom}(D_{\omega , \lambda }^{\min }). \end{aligned}$$
    (4.3)
  2. 2.

    If \(|\text {Re}(\mu )|<\frac{1}{2}\), then

    $$\begin{aligned} \dim \text {Dom}(D_{\omega , \lambda }^{\max })/\text {Dom}(D_{\omega , \lambda }^{\min })=2. \end{aligned}$$
    (4.4)

    Besides, if \(\chi \in C_\mathrm {c}^\infty ([0,\infty [)\) equals 1 near 0, then

    $$\begin{aligned} \text {Dom}(D_{\omega , \lambda }^{\max })=\text {Dom}( D_{\omega , \lambda }^{\min })+ \{f\chi \, | \, f\in \text {Ker}(D_{\omega , \lambda })\} .\end{aligned}$$
    (4.5)

We will prove the above theorem in the next section. Now we would like to discuss its consequences. If \(|\text {Re}(\mu )| < \frac{1}{2}\), we are especially interested in operators \(D_{\omega , \lambda }^\bullet \) satisfying

$$\begin{aligned} D_{\omega , \lambda }^{\min }\subsetneq D_{\omega , \lambda }^\bullet \subsetneq D_{\omega , \lambda }^{\max }. \end{aligned}$$
(4.6)

By the above theorem, they are in \(1-1\) correspondence with rays in \(\text {Ker}(D_{\omega , \lambda })\).

More precisely, let \(f\in \text {Ker}(D_{\omega , \lambda })\), \(f\ne 0\). Define \(D_{\omega , \lambda }^f\) as the restriction of \(D_{\omega , \lambda }^{\max }\) to

$$\begin{aligned} \text {Dom}(D_{\omega , \lambda }^f):=\text {Dom}(D_{\omega , \lambda }^{\min })+\mathbb {C}f\chi . \end{aligned}$$
(4.7)

Then \(D_{\omega , \lambda }^f\) is independent of the choice of \(\chi \) and satisfies

$$\begin{aligned} D_{\omega , \lambda }^{\min }\subsetneq D_{\omega , \lambda }^f\subsetneq D_{\omega , \lambda }^{\max }. \end{aligned}$$
(4.8)

Every \(D_{\omega , \lambda }^{\bullet }\) satisfying (4.6) is of the form \(D_{\omega , \lambda }^f\) for some f and we have \(D_{\omega , \lambda }^f = D_{\omega , \lambda }^g\) if and only if f and g are proportional to each other.

We will now investigate the domain of the minimal operator. Note that if we know the domain of \(D_{\omega , \lambda }^{\min }\), then the domain of \(D_{\omega , \lambda }^{\max }\) is also known from Theorem 13. From now on we do not use this result until its proof is presented.

The following two facts are well-known:

Lemma 14

Hardy’s inequality: If \( f \in H_0^1(\mathbb {R}_+)\), then

$$\begin{aligned} \int _{0}^{\infty } \frac{|f(x)|^2}{x^2} \text {d}x \le 4 \int _{0}^{\infty } |f'(x)|^2 \text {d}x. \end{aligned}$$
(4.9)

Lemma 15

If RS are closed operators such that R has bounded inverse, then RS is closed.

The above two lemmas are used in the following characterization of the minimal domain:

Proposition 16

\(H_0^1(\mathbb {R}_+, \mathbb {C}^2) \subset \text {Dom}(D_{\omega , \lambda }^{\min })\), with an equality if \(|\text {Re}(\mu )| \ne \frac{1}{2}\).

Proof

The inclusion follows from Hardy’s inequality. To prove the second part of the statement, we use Lemma 15. Consider \(R = A -M_{\omega , \lambda }\), \(S = \frac{\sigma _2}{x}\), where \(M_{\omega , \lambda } = \begin{bmatrix} \frac{\text {i}}{2} &{} - \text {i}\lambda -\text {i}\omega \\ \text {i}\lambda - \text {i}\omega &{} \frac{\text {i}}{2} \end{bmatrix}\). R is a bounded perturbation of A, so \(\text {Dom}(R)= \text {Dom}(A)\) and R is closed, while S is self-adjoint on the domain \(\text {Dom}(S) = \left\{ f \in L^2(\mathbb {R}_+,\mathbb {C}^2) \, | \, \frac{1}{x} f(x) \in L^2(\mathbb {R}_+,\mathbb {C}^2) \right\} \). One checks that \(RS = \left. D_{\omega , \lambda } \right| _{\text {Dom}(RS)}\). Next we show that \(\text {Dom}(RS) = H_0^1(\mathbb {R}_+,\mathbb {C}^2)\).

If \(f \in \text {Dom}(RS)\), then \(x \mapsto \frac{f(x)}{x}\) belongs to \( \text {Dom}(A)\). Since \(x \partial _x \frac{f(x)}{x} = f'(x) - \frac{f(x)}{x}\), this entails that \(f' \in L^2\). Thus, \(f \in H^1(\mathbb {R}_+, \mathbb {C}^2) \cap \text {Dom}(S) = H_0^1(\mathbb {R}_+,\mathbb {C}^2)\). Conversely, if \(f \in H_0^1(\mathbb {R}_+,\mathbb {C}^2)\), then \(f \in \text {Dom}(S)\) by Hardy’s inequality, while the last computation implies that \(Sf \in \text {Dom}(A)\). Thus, \(f \in \text {Dom}(RS)\).

We have shown that \(\text {Dom}(RS) = H_0^1(\mathbb {R}_+,\mathbb {C}^2)\), which is dense in \(\text {Dom}(D_{\omega , \lambda }^{\min })\) with the graph topology. Thus, \(D_{\omega , \lambda }^{\min }\) is the closure of RS. We have to check that R has bounded inverse.

If \(\mu \ne 0\), then \(M_{\omega , \lambda }\) is a diagonalizable matrix with eigenvalues \(c_{\pm }=\frac{\text {i}}{2} \pm \text {i}\mu \), which have nonzero imaginary part if \(|\text {Re}(\mu )| \ne \frac{1}{2}\). Therefore, the operator \(A- M_{\omega , \lambda }\) is similar to \(A - \begin{bmatrix} c_+ &{} 0 \\ 0 &{} c_- \end{bmatrix}\), which clearly is boundedly invertible. If \(\mu =0\), then \(N_{\omega ,\lambda }=M_{\omega , \lambda } - \frac{\text {i}}{2}\) is a nilpotent matrix, \(N_{\omega , \lambda }^2=0\). Therefore, \((A- M_{\omega , \lambda })^{-1} = (A - \frac{\text {i}}{2})^{-1} + (A - \frac{\text {i}}{2})^{-1} N_{\omega , \lambda } (A - \frac{\text {i}}{2})^{-1}\). \(\square \)

Corollary 17

\(D_{\omega , \lambda }^{\min }\) and \(D_{\omega , \lambda }^{\max }\) are holomorphic families of closed operators for \(|\text {Re}(\mu )| \ne \frac{1}{2}\).

Proof

Away from the set \(|\text {Re}(\mu )| = \frac{1}{2}\), the operators \(D_{\omega , \lambda }^{\min }\) have a constant domain. By Hardy’s inequality, \(D_{\omega , \lambda } f\) is a holomorphic family of elements of \(L^2(\mathbb {R}_+,\mathbb {C}^2)\) for any \(f \in H_0^1(\mathbb {R}_+,\mathbb {C}^2)\). Hence, \(D_{\omega , \lambda }^{\min }\) form a holomorphic family of bounded operators \(H_0^1(\mathbb {R}_+,\mathbb {C}^2) \rightarrow L^2(\mathbb {R}_+,\mathbb {C}^2)\). The claim for \(D_{\omega , \lambda }^{\max }\) follows by taking adjoints (see, e.g., Theorem 3.42 in [14]). \(\square \)

We denote by \(\sigma _\text {p}(B)\) the point spectrum of an operator B, that is

$$\begin{aligned} \sigma _\text {p}(B):=\{k\in \mathbb {C}\, | \, \dim (\text {Ker}(B-k)\ge 1\}. \end{aligned}$$
(4.10)

If \(\dim (\text {Ker}(B-k))=1\), we say that k is a nondegenerate eigenvalue.

In the following proposition we give a complete description of the point spectrum of the maximal and minimal operator. With no loss of generality, we assume that \(\text {Re}(\mu ) > - \frac{1}{2}\). Note that the definition of \(\mathcal {E}^\pm \) is not symmetric with respect to \(\mu \mapsto - \mu \)!

Proposition 18

One of the following mutually exclusive statements is true:

  1. 1.

    \(\text {Re}(\mu ) \ge \frac{1}{2}\) and \((\omega ,\lambda ,\mu )\in \mathcal {E}^\pm \). Then

    $$\begin{aligned}\sigma _\text {p}(D_{\omega ,\lambda }^{\max })=\sigma _\text {p}(D_{\omega ,\lambda }^{\min })=\mathbb {C}_\pm .\end{aligned}$$
  2. 2.

    \(\text {Re}(\mu ) \ge \frac{1}{2}\) and \((\omega ,\lambda ,\mu )\not \in \mathcal {E}^+\cup \mathcal {E}^-\). Then

    $$\begin{aligned}\sigma _\text {p}(D_{\omega ,\lambda }^{\max })=\sigma _\text {p}(D_{\omega ,\lambda }^{\min })=\emptyset .\end{aligned}$$
  3. 3.

    \(\text {Re}(\mu ) < \frac{1}{2}\) and \(|\text {Im}(\lambda )| \le \frac{1}{2}\). Then

    $$\begin{aligned}\sigma _\text {p}(D_{\omega ,\lambda }^{\max })=\mathbb {C}\backslash \mathbb {R},\qquad \sigma _\text {p}(D_{\omega ,\lambda }^{\min })=\emptyset .\end{aligned}$$
  4. 4.

    \(\text {Re}(\mu ) < \frac{1}{2}\) and \(|\text {Im}(\lambda )| > \frac{1}{2}\). Then

    $$\begin{aligned}\sigma _\text {p}(D_{\omega ,\lambda }^{\max })=\mathbb {C}^\times ,\qquad \sigma _\text {p}(D_{\omega ,\lambda }^{\min })=\emptyset .\end{aligned}$$

Besides, all eigenvalues of \(D_{\omega ,\lambda }^{\max }\) and \(D_{\omega ,\lambda }^{\min }\) are nondegenerate.

Proof

The four possibilities listed above are clearly mutually exclusive and cover all cases. Indeed, case \(p \in \mathcal {E}^+ \cap \mathcal {E}^-\) is ruled out by Lemma 4.

By Lemma 5, every \(f \in \text {Ker}(D_{\omega , \lambda }^{\max } -k)\) is a smooth function satisfying the differential equation \(( D_{\omega , \lambda }-k) f =0\), in which derivatives may be understood in the classical sense. Space of solutions of this equation is two-dimensional.

By discussion in Sect. 3, there exist no nonzero solutions in \(L^2(\mathbb {R}_+,\mathbb {C}^2)\) for \(k=0\). In the remainder of the proof we restrict attention to \(k \ne 0\).

First suppose that \(\text {Re}(\mu )\ge \frac{1}{2}\). If \(p \notin \mathcal {E}^+ \cup \mathcal {E}^-\), then \(\xi _p^+\) (as well as \(\xi _p^-\)) is the unique up to scalars solution square integrable in a neighborhood of zero, since other solutions have leading term proportional to \(x^{- \mu }\). It is not in \(L^2(\mathbb {R}_+,\mathbb {C}^2)\). Now let \(p \in \mathcal {E}^{\pm }\). If \(\pm \text {Im}(k) \le 0\), we can argue in the same way using function \(\xi _p^{\mp }\). In the case \(k \in \mathbb {C}_\pm \) solution \(\zeta _p^{\pm }\) is square integrable, whereas solutions not proportional to it grow exponentially at infinity. If \(\text {Re}(\mu ) > \frac{1}{2}\), then we have also \(\zeta _p^\pm \in H_0^1(\mathbb {R}_+,\mathbb {C}^2) \subset \text {Dom}(D_{\omega , \lambda }^{\min })\).

If \(\text {Re}(\mu ) = \frac{1}{2}\), then \(\zeta _p^{\pm } \notin H_0^1(\mathbb {R}_+,\mathbb {C}^2)\). We will now show that nevertheless \(\zeta _p^{\pm } \in \text {Dom}(D_{\omega , \lambda }^{\min })\). We define \(\zeta _{p,\epsilon }^\pm (k,x) = \min \{ x^{\epsilon } ,1 \} \zeta _p^\pm (k,x)\) for \(\epsilon >0\). Then \(\zeta _{p,\epsilon }^\pm \in H_0^1(\mathbb {R}_+,\mathbb {C}^2) \subset \text {Dom}(D_{\omega , \lambda }^{\min })\). We will show that \(\zeta _{p,\epsilon }^\pm \) converges to \(\zeta _p^\pm \) in the graph topology of \(\text {Dom}(D_{\omega , \lambda }^{\max })\) as \(\epsilon \rightarrow 0\). Indeed, convergence in \(L^2(\mathbb {R}_+,\mathbb {C}^2)\) is clear. Furthermore,

$$\begin{aligned} D_{\omega , \lambda }^{\min } \zeta _{p,\epsilon }^\pm (k,x) = k \zeta ^\pm _{p,\epsilon }(k,x) + \epsilon \, \mathbbm {1}_{[0,1]}(x) x^{\epsilon -1} \begin{bmatrix} 0 &{} -1 \\ 1 &{} 0 \end{bmatrix} \zeta _p^\pm (k,x), \end{aligned}$$
(4.11)

where \(\mathbbm {1}_{[0,1]}\) is the characteristic function of [0, 1]. The first term converges to \(k \zeta _p^\pm = D_{\omega , \lambda }^{\max } \zeta _p^\pm \). We show that the second term converges to zero by estimating

$$\begin{aligned}&\int _0^{\infty } \left| \epsilon \, \mathbbm {1}_{[0,1]}(x) x^{\epsilon -1} \begin{bmatrix} 0 &{} -1 \\ 1 &{} 0 \end{bmatrix} \zeta _p^\pm (k,x) \right| ^2 \text {d}x = \epsilon ^2 \int _0^1 \frac{|\zeta _p^\pm (k,x)|^2}{x} x^{2 \epsilon -1} \text {d}x \nonumber \\&\quad \le \epsilon ^2 \sup _{y \in [0,1]} \frac{|\zeta _p^\pm (k,y)|^2}{y} \cdot \int _0^1 x^{2 \epsilon -1} \text {d}x = \frac{\epsilon }{2} \sup _{y \in [0,1]} \frac{|\zeta _p^\pm (k,y)|^2}{y} . \end{aligned}$$
(4.12)

Now suppose that \(|\text {Re}(\mu ) |< \frac{1}{2}\). Then all solutions are square integrable in a neighborhood of the origin, but they do not belong to \(H_0^1(\mathbb {R}_+,C^2) = \text {Dom}(D_{\omega , \lambda }^{\min })\). If \(k \in \mathbb {C}_\pm \), then \(\zeta _p^\pm \) is square integrable and solutions not proportional to it grow at infinity.

It only remains to consider the case of nonzero \(k \in \mathbb {R}\). There exist solutions with leading terms for \(x \rightarrow \infty \) proportional to \(\text {e}^{- \text {i}k x} (kx)^{- \text {i}\lambda }\) and \(\text {e}^{\text {i}k x} (kx)^{ \text {i}\lambda }\). If \(|\text {Im}(\lambda )| > \frac{1}{2}\), then one of these two is square integrable. \(\square \)

We note that Proposition 18 partially describes also ranges of \(D_{\omega ,\lambda }^{\min }\) and \(D_{\omega ,\lambda }^{\max }\), since

$$\begin{aligned} \text {Ran}(D_{\omega , \lambda }^{\min }-k)^{\text {perp}} = \text {Ker}(D_{\omega , \lambda }^{\max }-k), \end{aligned}$$
(4.13)
$$\begin{aligned} \text {Ran}(D_{\omega , \lambda }^{\max }-k)^{\text {perp}} = \text {Ker}(D_{\omega , \lambda }^{\min }-k), \end{aligned}$$
(4.14)

Corollary 19

Operators \(D_{\omega , \lambda }^{\begin{array}{c} \min \end{array}}\) and \(D_{\omega , \lambda }^{\begin{array}{c} \max \end{array}}\) have empty resolvent sets if \(|\text {Re}(\mu )| < \frac{1}{2}\).

Homogeneous Realizations and the Resolvent

Definition and Basic Properties

We consider the following open subset of \(\mathcal {M}\):

$$\begin{aligned} \mathcal {M}_{-\frac{1}{2}}:=\left\{ p \in \mathcal {M}\ | \ \text {Re}(\mu ) > - \frac{1}{2} \right\} . \end{aligned}$$
(5.1)

As before, choose \(\chi \in C_\mathrm {c}^\infty (\mathbb {R}_+)\) equal to 1 near 0. If \(p = (\omega , \lambda , \mu ,[a:b]) \in \mathcal {M}_{- \frac{1}{2}}\), we define \(D_p\) to be the restriction of \(D_{\omega ,\lambda }^{\max }\) to

$$\begin{aligned} \text {Dom}(D_p):=\text {Dom}(D_{\omega ,\lambda }^{\min })+\mathbb {C}x^\mu \begin{bmatrix} a \\ b \end{bmatrix} \chi . \end{aligned}$$
(5.2)

This definition is correct because \(x^{\mu } \begin{bmatrix} a \\ b \end{bmatrix}\chi \) is an element of \(\text {Dom}(D_{\omega , \lambda }^{\max })\) for \(\text {Re}(\mu ) > - \frac{1}{2}\). If \(\text {Re}(\mu ) \ge \frac{1}{2}\), then it belongs to \(\text {Dom}(D_{\omega , \lambda }^{\min })\), so we have \(D_p = D_{\omega , \lambda }^{\min }\).

Theorem 20

Let \(p\in \mathcal {M}_{{}-\frac{1}{2}}\). Then the operator \(D_p\) does not depend on the choice of \(\chi \), is closed, self-transposed and

$$\begin{aligned}&\sigma (D_p) = {\left\{ \begin{array}{ll} \mathbb {R}&{} \text {for } p \notin \mathcal {E}^+ \cup \mathcal {E}^-, \\ \mathbb {C}_\pm \cup \mathbb {R}&{} \text {for } p \in \mathcal {E}^{\pm }, \end{array}\right. } \nonumber \\&\quad \sigma _\text {p}(D_p) = {\left\{ \begin{array}{ll} \emptyset &{} \text {for } p \notin \mathcal {E}^+ \cup \mathcal {E}^-, \\ \mathbb {C}_\pm &{} \text {for } p \in \mathcal {E}^{\pm }. \end{array}\right. } \end{aligned}$$
(5.3)

If \(\pm \text {Im}(k) >0\) and \(p\not \in \mathcal {E}^{\pm }\), then the integral kernel \(G_p^{\bowtie }(k;x,y)\) introduced in (3.26) defines a bounded operator \(G_p^{\bowtie }(k)\) and

$$\begin{aligned} G_p^{\bowtie }(k)=(D_p-k)^{-1}. \end{aligned}$$
(5.4)

For \(k\in \mathbb {C}_\pm \), the map \(\mathcal {M}_{{}-\frac{1}{2}} \backslash \mathcal {E}^\pm \ni p\mapsto (D_p-k)^{-1}\) is a holomorphic family of bounded operators.

Therefore, \(\mathcal {M}_{{}-\frac{1}{2}} \ni p\mapsto D_p\) is a holomorphic family of closed operators.

Proof

It is sufficient to consider the case \(\text {Im}(k)<0\). Let \(p \notin \mathcal {E}^-\). We prove the boundedness separately for the integral operators with kernels \(G^{\bowtie }_{p}(k)\) restricted to four regions forming a partition of \(\mathbb {R}_+ \times \mathbb {R}_+\) (up to an inconsequential overlap on a set of measure zero). Throughout the proof we use notation \(x_< = \min \{ x, y \}\), \(x_> = \max \{ x, y \}\). Symbols \(c_{p}, c_{p}'\) will be used for positive constants which are locally bounded functions of p.

First we consider the region \(x,y \le |k|^{-1}\). Inspecting the asymptotics of Whittaker functions for small argument we conclude that \(|G^{\bowtie }_{p}(k;x,y)| \le c_{p} (|k|x_<)^{\text {Re}(\mu )} (|k| x_>)^{-|\text {Re}(\mu )|}\). Using this inequality and elementary integrals, we estimate

$$\begin{aligned} \int _{\left[ 0, |k|^{-1} \right] ^2} |G^{\bowtie }_{p}(k;x,y)|^2 \text {d}x \text {d}y \le \frac{c_{p}^2}{|k|^2} \frac{1}{1 + 2 \text {Re}(\mu )} \frac{1}{1+ \text {Re}(\mu ) - |\text {Re}(\mu )|}. \end{aligned}$$
(5.5)

Therefore, the Hilbert–Schmidt norm of the corresponding operator is bounded by \(\frac{c_p'}{|k|}\).

Next, in the region \(y \le |k|^{-1} \le x\) we have \(|G^{\bowtie }_{p}(k)| \le c_{p} (|k| y)^{\text {Re}(\mu )} (|k|x)^{\text {Im}(\lambda )} \text {e}^{-|\text {Im}(k)| x} \). Thus,

$$\begin{aligned}&\int \limits _{[|k|^{-1}, \infty [ \times [0,|k|^{-1}]} |G^{\bowtie }_{p}(k;x,y)|^2 \text {d}x \text {d}y \nonumber \\&\quad \le \frac{c_{p}^2}{|k|^2} \int _{[1, \infty [ \times [0,1]} \text {e}^{- 2 \frac{|\text {Im}(k)|}{|k|} t} t^{2 \, \text {Im}(\lambda )} t'^{2 \text {Re}(\mu )} \text {d}t \text {d}t', \end{aligned}$$
(5.6)

which is a convergent integral depending continuously on \(\lambda , \mu \). Again, the corresponding operator is Hilbert–Schmidt with locally bounded norm. By the symmetry property (3.27) the same is true for the region \(x \le |k|^{-1} \le y\).

Finally for \(x, y \ge |k|^{-1}\) we have \(|G^{\bowtie }_{p}(k;x,y)| \le c_{p} \text {e}^{-|\text {Im}(k)| (x_> - x_<)} \frac{x_>^{\text {Im}(\lambda )}}{x_<^{\text {Im}(\lambda )}}\). Hence,

$$\begin{aligned}&\int _{|k|^{-1}}^{\infty } |G^{\bowtie }_{p} (k;x,y)| \text {d}y \nonumber \\&\quad \le c_{p} \left( \int _{|k|^{-1}}^x \text {e}^{ |\text {Im}(k)| (y-x)} \frac{y^{- \text {Im}(\lambda )}}{x^{- \text {Im}(\lambda )}} \text {d}y + \int _x^{\infty } \text {e}^{-|\text {Im}(k)| (y-x)} \frac{y^{\text {Im}(\lambda )}}{x^{\text {Im}(\lambda )}} \text {d}y \right) . \end{aligned}$$
(5.7)

If \(\text {Im}(\lambda ) \le 0\), then \( \frac{y^{{\mp } \text {Im}(\lambda )}}{x^{{\mp } \text {Im}(\lambda )}} \le 1\) under these integrals, so elementary calculation gives

$$\begin{aligned} \int _{|k|^{-1}}^{\infty } |G^{\bowtie }_{p} (k;x,y)| \text {d}y \le \frac{2 c_p}{|\text {Im}(k)|}. \end{aligned}$$
(5.8)

Next we consider the case \(\text {Im}(\lambda ) >0\). Integration by parts in the first term of (5.7) gives

$$\begin{aligned}&\int _{|k|^{-1}}^x \text {e}^{ |\text {Im}(k)| (y-x)} \frac{y^{- \text {Im}(\lambda )}}{x^{- \text {Im}(\lambda )}} \text {d}y \le \frac{1}{|\text {Im}(k)|} \nonumber \\&\quad +\, \frac{\text {Im}(\lambda )}{|\text {Im}(k)|} \int _{|k|^{-1}}^x \text {e}^{|\text {Im}(k)|(y-x)} \frac{x^{\text {Im}(\lambda )}}{y^{\text {Im}(\lambda )+1}} \text {d}y . \end{aligned}$$
(5.9)

The integrand of this integral is maximized at one of the two endpoints, so

$$\begin{aligned} \int _{|k|^{-1}}^x \text {e}^{|\text {Im}(k)|(y-x)} \frac{x^{\text {Im}(\lambda )}}{y^{\text {Im}(\lambda )+1}} \text {d}y&\le \max \left\{ \text {e}^{\frac{|\text {Im}(k)|}{|k|}(1-|k|x) } \frac{(|k|x)^{\text {Im}(\lambda )+1}}{x} , \frac{1}{x} \right\} \int _{|k|^{-1}}^x \text {d}y \nonumber \\&\le \max \left\{ \text {e}^{1 - |\text {Im}(k)|x } (|k|x)^{\text {Im}(\lambda )+1} , 1 \right\} . \end{aligned}$$
(5.10)

Optimizing with respect to x we conclude that

$$\begin{aligned} \int _{|k|^{-1}}^x \text {e}^{|\text {Im}(k)|(y-x)} \frac{x^{\text {Im}(\lambda )}}{y^{\text {Im}(\lambda )+1}} \text {d}y \le \max \left\{ \text {e}\left( \frac{\text {Im}(\lambda )+1}{|\text {Im}(k)|} \right) ^{\text {Im}(\lambda )+1} , 1 \right\} . \end{aligned}$$
(5.11)

In the second integral in (5.7), we integrate by parts \(n \ge \text {Im}(\lambda )\) times:

$$\begin{aligned} \int _x^{\infty } \text {e}^{-|\text {Im}(k)| (y-x)} \frac{y^{\text {Im}(\lambda )}}{x^{\text {Im}(\lambda )}} \text {d}y= & {} \sum _{j=0}^{n-1} \frac{c_j x^{-j}}{| \text {Im}(k)|^{j+1}} \nonumber \\&\quad + \frac{c_n}{|\text {Im}(k)|^{n}} \int _{y}^{\infty } \text {e}^{-|\text {Im}(k)| (y-x)} \frac{y^{\text {Im}(\lambda )-n}}{x^{\text {Im}(\lambda )}} \text {d}y ,\nonumber \\ \end{aligned}$$
(5.12)

where \(c_j:=\text {Im}(\lambda ) (\text {Im}(\lambda )-1)\cdots (\text {Im}(\lambda )-j+1)\). Next we estimate \(y^{\text {Im}(\lambda )-n} \le x^{\text {Im}(\lambda )-n}\) and \(x^{-j} \le |k|^j\) under the remaining integral. Then simple calculation gives

$$\begin{aligned} \int _x^{\infty } \text {e}^{-|\text {Im}(k)| (y-x)} \frac{y^{\text {Im}(\lambda )}}{x^{\text {Im}(\lambda )}} \text {d}y \le \sum _{j=0}^{n} \frac{\text {Im}(\lambda )_j |k|^j}{| \text {Im}(k)|^{j+1}}. \end{aligned}$$
(5.13)

The same estimates are true for \(\int _{|k|^{-1}}^{\infty } |G^{\bowtie }_{p}(k;x,y)|\text {d}x\). The claim follows by Schur’s criterion. This proves the boundedness of \(G^{\bowtie }_p(k)\).

Equation (3.27) implies that (whenever \(G^{\bowtie }_{p}(k)\) is defined) we have \(\langle f|G^{\bowtie }_{p}(k)g\rangle = \langle G^{\bowtie }_{p}(k) f|g\rangle \) for \(f,g \in C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\). By continuity, the same is true for all \(f,g \in L^2(\mathbb {R}_+,\mathbb {C}^2)\). Thus, \(G^{\bowtie }_{p}(k)\) is self-transposed.

Next we check that \(\langle f|D_p g\rangle = \langle D_p f|g\rangle \) for \(f,g \in \text {Dom}(D_p)\). To this end, we evaluate

$$\begin{aligned} \langle f|D_p g\rangle - \langle D_p f |g\rangle = \text {i}\int _0^{\infty } \frac{\text {d}}{\text {d}x} \left( f(x)^\text {T}\sigma _2 g(x) \right) \text {d}x. \end{aligned}$$
(5.14)

If either f or g is in \(C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\), the right-hand side is zero. By continuity with respect to the graph norm, the same is true for all \(f,g \in \text {Dom}(D_{\omega , \lambda }^{\min })\). Since \(\sigma _2\) is a skew-symmetric matrix, the right-hand side vanishes also for fg proportional to \(\chi x^{\mu } \begin{bmatrix} a \\ b \end{bmatrix}\). Thus, \(D_p\) is self-transposed.

Let \(f \in L^2(\mathbb {R}_+,\mathbb {C}^2)\). We pick a sequence \(f_i \in C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\) such that \(f_i \rightarrow f\). Then \(G^{\bowtie }_{p}(k) f_i \rightarrow G^{\bowtie }_{p}(k) f\) and

$$\begin{aligned} D_{p}G^{\bowtie }_{p}(k) f_i =f_i + k G^{\bowtie }_{p}(k) f_i \rightarrow f + k G^{\bowtie }_{p}(k) f. \end{aligned}$$
(5.15)

Since \(D_{p}\) is closed, this implies that \(f \in \text {Dom}(D_{p})\) and \(D_{p}G^{\bowtie }_{p}(k) f = f + k G^{\bowtie }_{p}(k) f\). Therefore, we have \(\text {Ran}(G^{\bowtie }_{p}(k)) \subset \text {Dom}(D_p)\) and \((D_{p}-k)G^{\bowtie }_{p}(k)=1\).

For any \(f \in L^2(\mathbb {R}_+,\mathbb {C}^2)\) and \(g \in \text {Dom}(D_{p})\) we have

$$\begin{aligned}&\langle f| G^{\bowtie }_{p}(k) (D_p -k)g\rangle = \langle G^{\bowtie }_{p}(k) f|(D_p-k)g\rangle \nonumber \\&\quad =\langle (D_p-k)G^{\bowtie }_{p}(k)f|g\rangle =\langle f|g\rangle . \end{aligned}$$
(5.16)

Since f was arbitrary, \(G^{\bowtie }_{p}(k) (D_{p}-k) g = g\). Thus, \(k \notin \sigma (D_p)\) and \(G^{\bowtie }_{p}(k)=(D_{p}-k)^{-1}\).

To show that \((D_p-k)^{-1}\) is unbounded for \(k \in {\mathbb {R}}^\times \), we fix \(\epsilon >0\) and consider the function

$$\begin{aligned} f_{\epsilon }(x) = \text {e}^{- \epsilon x} \xi _{p}( k , x). \end{aligned}$$

Then \(f_{\epsilon } \in \text {Dom}(D_{p})\) and \(|(D_{p}-k) f_{\epsilon } (x)|= \epsilon |f_{\epsilon } (x)|\), so \(\frac{\Vert (D_{p}-k) f_{\epsilon } \Vert }{\Vert f_{\epsilon } \Vert } = \epsilon \). Hence, \(k \in \sigma (D_p)\). Since \(\sigma (D_p)\) is closed, \(\mathbb {R}\subset \sigma (D_p)\).

Finally, let \(p \in \mathcal {E}^\pm \), \(k \in \mathbb {C}_\pm \). Then \(\zeta _p^\pm (k, \cdot )\) belongs to \(\text {Ker}(D_p - k)\). \(\square \)

Corollary 21

We have \(D_p^* = D_{{\overline{p}}}\). In particular \(D_p\) is self-adjoint if \(p = {\overline{p}}\).

We are now ready to prove Theorem 13.

Proof of Theorem 13

We choose some k in the resolvent set of \(D_p\).

If \(|\text {Re}(\mu )| \ge \frac{1}{2}\), then \(D_{\omega , \lambda }^{\min } =D_p\), so it suffices to show that \(D_p = D_{\omega , \lambda }^{\max }\). Indeed, \(D_p - k\) is surjective, so the ranges of \(D_p-k\) and \(D_{\omega ,\lambda }^{\max }-k\) coincide. By Proposition 18 also kernels are equal.

Next we consider the case \(|\text {Re}(\mu )| < \frac{1}{2}\).

We easily check that \(\chi x^{\mu } \begin{bmatrix} a \\ b \end{bmatrix} \notin H_0^1(\mathbb {R}_+,\mathbb {C}^2)\), which by Proposition 16 for \(|\text {Re}(\mu )| < \frac{1}{2}\) coincides with \(\text {Dom}(D_{\omega , \lambda }^{\min })\). Hence, \(\text {Dom}(D_{\omega , \lambda }^{\min })\) is a codimension one subspace of \(\text {Dom}(D_p)\).

Next, \(D_{p}-k\) and \(D_{\omega , \lambda }^{\max }-k\) have the same range–the whole Hilbert space. Besides, \(\dim \text {Ker}(D_{\omega , \lambda }^{\max }-k)=1\) by Proposition 18. Hence, \(\text {Dom}(D_p)\) is a codimension one subspace of \(\text {Dom}(D_{\omega , \lambda }^{\max })\). \(\square \)

Proposition 22

Family \(D_p\) has the following symmetries

$$\begin{aligned} \sigma _1 D_{\omega , \lambda , \mu , [a:b]} \sigma _1&= - D_{\omega , - \lambda , \mu ,[b:a]}, \end{aligned}$$
(5.17a)
$$\begin{aligned} \sigma _2 D_{\omega , \lambda , \mu , [a:b]} \sigma _2&= D_{- \omega , \lambda , \mu , [-b:a]}, \end{aligned}$$
(5.17b)
$$\begin{aligned} \sigma _3 D_{\omega , \lambda , \mu , [a:b]} \sigma _3&= -D_{-\omega , - \lambda , [-a:b]}, \end{aligned}$$
(5.17c)

where \(\sigma _j\) are the Pauli matrices.

Proof

Matrix multiplication gives

$$\begin{aligned}&\sigma _1 D_{\omega , \lambda } \sigma _1 = - D_{\omega , - \lambda }, \qquad \sigma _2 D_{\omega , \lambda } \sigma _2 = D_{- \omega , \lambda }, \nonumber \\&\quad \sigma _3 D_{\omega , \lambda } \sigma _3 =-D_{- \omega , - \lambda }. \end{aligned}$$
(5.18)

Using (5.2) one checks that the domains of operators on the left and right-hand side of (22) agree. \(\square \)

Essential Spectrum

Proposition 23

Let \(p,p' \in \mathcal {M}_{{}-\frac{1}{2}}\) and \(k\not \in \sigma (D_p)\cup \sigma (D_{p'})\). Then \(G^{\bowtie }_{p}(k)-G^{\bowtie }_{p'}(k)\) is a Hilbert–Schmidt operator.

Proof

The proof of Theorem 20 shows that it suffices to show that the integral operator with kernel \(G^{\bowtie }_{p}(k) - G^{\bowtie }_{p'}(k)\) restricted to the region \(x,y \ge |k|^{-1}\) is Hilbert–Schmidt. Furthermore, we may assume that \(\text {Im}(k)<0\). Using formulas (3.18) and (3.19), we obtain the following asymptotic expansion for \(x, y \rightarrow \infty \):

$$\begin{aligned} - \xi _{p}^-(k,x) \otimes \zeta _{p}^-(k,y) \sim \frac{1}{2 \text {i}} \begin{bmatrix} 1 \\ \text {i}\end{bmatrix} \otimes \begin{bmatrix} 1 \\ - \text {i}\end{bmatrix} \cdot \left( \frac{x}{y } \right) ^{ \text {i}\lambda } \text {e}^{\text {i}k(x-y )}. \end{aligned}$$
(5.19)

It follows that we have

$$\begin{aligned}&|G^{\bowtie }_{p}(k;x,y ) - G^{\bowtie }_{p'}(k;x,y )| \nonumber \\&\quad \le c \, \text {e}^{- |\text {Im}(k)| (x_> - x_<)} \left| \left( \frac{x_<}{x_>} \right) ^{ \text {i}\lambda } - \left( \frac{x_<}{x_>} \right) ^{ \text {i}\lambda '} \right| , \end{aligned}$$
(5.20)

with some constant c independent of xy. Therefore,

$$\begin{aligned} I&:= \int _{[|k|^{-1}, \infty ]^2} |G^{\bowtie }_{p}(k;x,y ) - G^{\bowtie }_{p'}(k;x,y )|^2 \text {d}x \text {d}y \nonumber \\&\le 2c^2 \int _{|k|^{-1}}^{\infty } \int _{y }^{\infty } \text {e}^{- 2 \, \text {Im}(z) (x-y )} \left| \left( \frac{x}{y } \right) ^{ -\text {i}\lambda } - \left( \frac{x}{y } \right) ^{- \text {i}\lambda '} \right| ^2 \text {d}x \text {d}y . \end{aligned}$$
(5.21)

Next we change variables to yt with \(x = t y \). This gives

$$\begin{aligned} I&\le \int _1^{\infty } \int _{|k|^{-1}}^{\infty } y \text {e}^{- 2 | \text {Im}(k)| (t-1) y } |t^{-\text {i}\lambda } - t^{-\text {i}\lambda '}|^2 \text {d}y \text {d}t \nonumber \\&= \int _1^{\infty } \frac{|k|+2 |\text {Im}(k)|(t-1)}{4 |\text {Im}(k)|^2 |k| (t-1)^2} \text {e}^{- 2 \frac{|\text {Im}(k)|}{|k|}(t-1)} |t^{-\text {i}\lambda } - t^{-\text {i}\lambda '}|^2 \text {d}t, \end{aligned}$$
(5.22)

where we have computed an elementary integral over y. The remaining integrand is bounded for \(t \rightarrow 1\) and decays exponentially for \(t \rightarrow \infty \). Therefore, the integral converges. \(\square \)

Resolvents of operators \(D_p\) for distinct \(p\in \mathcal {M}_{-\frac{1}{2}}\) are close to each other in the sense specified by Proposition 23. Therefore, it is useful to know that for some p their integral kernels are particularly simple. These are provided in the “Appendix A.3”.

By the essential spectrum (resp. essential spectrum of index zero) of a closed operator R, we mean the set \(\sigma _{\text {ess}}(R)\) (resp. \(\sigma _{\text {ess},0}(R)\)) of all \(k \in {\mathbb {C}}\) such that \(R-k\) is not a Fredholm operator (resp. Fredholm operator of index zero). Clearly \(\sigma _{\text {ess}}(R) \subset \sigma _{\text {ess},0}(R)\).

Lemma 24

Let RS be closed operators such that there exists \(k_0\) in the intersection of resolvent sets of R and S such that \((R-k_0)^{-1}-(S-k_0)^{-1}\) is a compact operator. Then \(\sigma _{\text {ess}}(R) = \sigma _{\text {ess}}(S)\) and \(\sigma _{\text {ess},0}(R) = \sigma _{\text {ess},0}(S)\).

Proof

By assumption, \((S-k_0)^{-1}\) and \((R-k_0)^{-1}\) have the same essential spectra. The spectral mapping theorem proven in [4] gives

$$\begin{aligned} \sigma _{\mathrm {ess}}(S)&= \{ k \in {\mathbb {C}} \, | \, \exists q \in \sigma _{\mathrm {ess}}((S-k_0)^{-1}) \ (k-k_0)q =1 \} \nonumber \\&= \{ k \in {\mathbb {C}} \, | \, \exists q \in \sigma _{\mathrm {ess}}((R-k_0)^{-1}) \ (k-k_0)q =1 \} = \sigma _{\mathrm {ess}}(R). \end{aligned}$$
(5.23)

The same argument works also for \(\sigma _{\text {ess},0}\). \(\square \)

Corollary 25

For any \(p\in \mathcal M_{{}-\frac{1}{2}}\) we have \(\sigma _{\text {ess}}(D_p)=\sigma _{\text {ess},0}(D_p) = {\mathbb {R}}\).

Proof

There exists p such that \(\sigma (D_p)=\mathbb {R}\). By Lemma 24, it is sufficient to prove our statement for such p. Clearly, \(\sigma _{\text {ess}}(D_p) \subset \sigma _{\text {ess},0}(D_p) \subset \sigma (D_p) = {\mathbb {R}}\). If \(k \in \mathbb {R}\), then \(D_p -k\) is injective and its range is dense, hence not closed, for otherwise \((D_p-k)^{-1}\) would be bounded. \(\square \)

Corollary 26

Let \(\omega , \lambda \) be such that \(|\text {Re}\sqrt{\omega ^2 - \lambda ^2}| < \frac{1}{2}\). Then \(\sigma _{\text {ess}}(D_{\omega ,\lambda }^{\min }) = \sigma _{\text {ess}}(D_{\omega ,\lambda }^{\max }) = \mathbb {R}\). If \(k \in \mathbb {C}{\setminus } \mathbb {R}\), then \(D_{\omega ,\lambda }^{\min } - k\) and \(D_{\omega ,\lambda }^{\max } - k\) are Fredholm operators with indices \(-1\) and \(+1\), respectively. If D is an operator satisfying \(D_{\omega ,\lambda }^{\min } \subsetneq D \subsetneq D_{\omega ,\lambda }^{\max }\), then \(\sigma _{\text {ess}}(D) = \sigma _{\text {ess},0}(D)=\mathbb {R}\).

Proof

Follows from Theorem 13 and Corollary 25. \(\square \)

Limiting Absorption Principle

Let \(s\in \mathbb {R}\). The Hilbert space \(L^2_s(\mathbb {R}_+,\mathbb {C}^2)\) is defined as the completion of \(C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\) with respect to the norm induced by the scalar product \(( f | g )_s = \int _0^{\infty } (1+x^2)^{s} \overline{f(x)} g(x) \text {d}x\). For any \(t \in {\mathbb {R}}\) we have a unitary operator \(\langle X \rangle ^t : L^2_s(\mathbb {R}_+,\mathbb {C}^2) \rightarrow L^2_{s-t}(\mathbb {R}_+,\mathbb {C}^2)\) given by \((\langle X \rangle ^t f)(x) = (1+x^2)^{\frac{t}{2}} f(x)\), alternatively regarded as an (unbounded for \(t>0\)) positive operator on \(L^2(\mathbb {R}_+,\mathbb {C}^2)\).

Proposition 27

Let \(p \in \mathcal {M}_{- \frac{1}{2}} \backslash \mathcal {E}^{\pm }\), \(k \in \mathbb {R}^\times \). The limit \(G^{\bowtie }_{p}(k \pm \text {i}0) \!:= \!\lim \limits _{\epsilon \downarrow 0} G^{\bowtie }_{p}( k \pm \text {i}\epsilon )\) exists as a compact operator \(L^2_s(\mathbb {R}_+,\mathbb {C}^2) \rightarrow L^2_{-s}(\mathbb {R}_+,\mathbb {C}^2)\) for any \(s > |\mathrm {Im}(\lambda )| + \frac{1}{2}\) and depends continuously on pk.

If \(\text {Re}(\mu ) >0\), then \(\mathbb {R}^\times \) may be replaced by \(\mathbb {R}\) in the above statement and \(G_p^{\bowtie }(\pm \text {i}0)\) has the kernel

$$\begin{aligned}&G_p^{\bowtie }(0;x,y) = \frac{1}{2} \mathbbm {1}_{\mathbb {R}_+}(y-x) \left( \frac{x}{y} \right) ^{\mu } \begin{bmatrix} z &{} -1 \\ 1 &{} - z^{-1} \end{bmatrix} \nonumber \\&\quad + \frac{1}{2} \mathbbm {1}_{\mathbb {R}_+}(x-y) \left( \frac{y}{x} \right) ^{\mu } \begin{bmatrix} z &{} 1 \\ -1 &{} - z^{-1} \end{bmatrix}. \end{aligned}$$
(5.24)

If \(\text {Re}(\mu ) \le 0\), then \(\Vert G_p^{\bowtie }(k) \Vert _{B(L^2_s,L^2_{-s})} = O(|k|^{2 \text {Re}(\mu )})\) for \(k \rightarrow 0\).

Therefore, in both cases we have \(\Vert G_p^{\bowtie }(\cdot \pm \text {i}0) \Vert _{B(L^2_s, L^2_{-s})} \in L^1_{\mathrm {loc}}(\mathbb {R})\).

Proof

It is sufficient to cover the case of k approaching the real axis from below. Asymptotics of \(G^{\bowtie }(k;x,y)\) are such that \((1+x^2)^{- \frac{s}{2}} (1+y ^2)^{- \frac{s}{2}} G^{\bowtie }_{p}(k;x,y )\) is an \(L^2({\mathbb {R}}_+^2 , \mathrm {End}(\mathbb {C}^2))\) function. Dominated convergence theorem implies that it depends continuously (in the \(L^2\) sense) on pk, including the boundary set \(\text {Im}(k)=0\). Therefore, \(\langle X \rangle ^{-s} G^{\bowtie }_{p}(k) \langle X \rangle ^{-s}\) is a continuous family of Hilbert–Schmidt (and hence compact) operators on \(L^2(\mathbb {R}_+,\mathbb {C}^2)\), so \(G^{\bowtie }_{p}(k)\) defines an operator \(L^2_s(\mathbb {R}_+,\mathbb {C}^2) \rightarrow L^2_{-s}(\mathbb {R}_+,\mathbb {C}^2)\) which may be written as a composition of two unitaries and a compact operator.

The second part follows from the asymptotics of \(\xi ^{\pm }_p\) and \(\zeta ^{\pm }_p\) functions for small arguments and the dominated convergence theorem. \(\square \)

Generalized Eigenvectors

Point spectrum of \(D_p\), when present, possesses quite counter-intuitive properties. Note that in this subsection an important role is played by the bilinear product \(\langle \cdot |\cdot \rangle \).

Proposition 28

Let \(n,m\in \mathbb {N}\). If \(f \in \text {Ker}((D_p-k)^n)\), \(g \in \text {Ker}((D_p-k')^m)\), then \(\langle f | g \rangle =0\).

Proof

Assume at first that \(k' \ne k\). We induct on m. If \(m=1\), then

$$\begin{aligned} 0= & {} \langle (D_p - k)^n f | g \rangle = \sum _{j=0}^n \left( {\begin{array}{c}n\\ j\end{array}}\right) (k'-k)^{n-j} \langle f | (D_p - k')^j g \rangle \nonumber \\= & {} (k'-k)^n \langle f | g \rangle . \end{aligned}$$
(5.25)

Cancelling \((k'-k)^n\) we obtain the induction base. Assume that the claim is true for m and let \(g \in \text {Ker}((D_p-k')^{m+1})\). By a similar calculation

$$\begin{aligned} 0= \sum _{j=0}^n \left( {\begin{array}{c}n\\ j\end{array}}\right) (k'-k)^{n-j} \langle f | (D_p - k')^j g \rangle = (k-k')^n \langle f | g \rangle , \end{aligned}$$
(5.26)

where the last equality follows from \((D_p-k')^j g \in \text {Ker}((D_p-k')^m)\) for \(j \ge 1\) and the induction hypothesis. This completes the proof for \(k \ne k'\).

So far we used only the self-transposedness of \(D_p\). Next we will also use its homogeneity.

Let \(k'=k\). Then for any \(\tau \in \mathbb {R}^{\times }\) we have \(U_{\tau } g \in \text {Ker}((D_p-k'')^m)\) for some \(k'' \ne k\). Hence, \(\langle f | U_{\tau } g \rangle =0\). Now take \(\tau \rightarrow 0\). \(\square \)

Proposition 29

If \(p \in \mathcal {E}^\pm \) and \(k \in \mathbb {C}_\pm \), then for every \(n \in {\mathbb {N}}\) we have \(\dim (\text {Ker}((D_p-k)^n)) = n\).

Proof

We proceed by induction on n. Case \(n=0\) is trivial and \(n=1\) is already established. By the inductive hypothesis, there exists \(f \in \text {Ker}((D_p-k)^n) \backslash \text {Ker}((D_p-k)^{n-1})\), unique up to elements of \(\text {Ker}((D_p-k)^{n-1})\) and multiplication by nonzero scalars. Then \(f \in \text {Ker}(D_p -k)^{\text {perp}}\) by Proposition 28. On the other hand, \(\text {Ker}(D_p-k)^{\text {perp}} = \left( \text {Ran}(D_p-k)^{\text {perp}} \right) ^{\text {perp}} = \text {Ran}(D_p - k)\). Here the last equality holds because \(D_p -k\) has closed range, see Corollary 25. Thus, there exists \(g \in \text {Dom}(D_p -k)\), unique up to elements of \(\text {Ker}(D_p - k )\), such that \((D_p-k) g =f\). Clearly, \(g \in \text {Ker}((D_p-k)^{n+1}) \backslash \text {Ker}((D_p-k)^n)\) and we have a vector space decomposition \(\text {Ker}((D_p-k)^{n+1}) = \mathbb {C}g \oplus \text {Ker}((D_p-k)^{n})\) . \(\square \)

Question 1

Let \(p \in \mathcal {E}^\pm \), \(k \in \mathbb {C}_{\pm }\). We denote the \(L^2\) closure of \(\bigcup \limits _{n=0}^{\infty } \text {Ker}((D_p - k)^n)\) by \({\mathcal {N}}_p(k)\). By Lemma 28 we have \({\mathcal {N}}_p(k) \subset {\mathcal {N}}_p(k)^\text {perp}\). In “Appendix A.3” we have verified that in the case \(\omega =0\) subspace \({\mathcal {N}}_p(k)\) does not depend on the choice of \(k \in \mathbb {C}_\pm \) and \({\mathcal {N}}_p(k) = {\mathcal {N}}_p(k)^\text {perp}\) (equivalently, \({\mathcal {N}}_p(k) \oplus \overline{{\mathcal {N}}_p(k)} = L^2(\mathbb {R}_+,\mathbb {C}^2)\)). We leave open the question whether these assertions remain true for \(\omega \ne 0\).

Diagonalization

Let \(k \in \mathbb {R}^\times \). Recall that \(\varepsilon _k = \text {sgn}(\text {Re}(k))\). On the real line, it is convenient to rewrite the formulas for \(\xi ^\pm \) and \(\zeta ^\pm \) (3.13, 3.15) in terms of trigonometric Whittaker functions (D.28, D.31):

$$\begin{aligned} \xi _p^{\pm }(k,x)= & {} \frac{\text {i}^{{\mp } \varepsilon _k \mu }}{2 N_p^{\pm } \mu } \left( \varepsilon _k \omega \mathcal {J}_{\varepsilon _k \lambda , \mu + \frac{1}{2}}(2|k|x) \begin{bmatrix} -z \\ 1 \end{bmatrix} \right. \nonumber \\&\quad \left. +\, \mathcal {J}_{\varepsilon _k \lambda , \mu - \frac{1}{2}} (2|k|x) \begin{bmatrix} z \\ 1 \end{bmatrix} \right) , \end{aligned}$$
(6.1a)
$$\begin{aligned} \zeta _p^{\pm }(k,x)= & {} \frac{\text {i}^{\pm \varepsilon _k \mu }}{2} \left( \pm \text {i}\varepsilon _k (z \pm \text {i}) \mathcal {H}^{\pm \varepsilon _k}_{\varepsilon _k \lambda , \mu + \frac{1}{2}} (2 |k| x) \begin{bmatrix} -1 \\ z^{-1} \end{bmatrix} \right. \nonumber \\&\quad \left. +\, (z {\mp } \text {i}) \mathcal {H}^{\pm \varepsilon _k}_{\varepsilon _k \lambda , \mu - \frac{1}{2}} (2 |k| x) \begin{bmatrix} 1 \\ z^{-1} \end{bmatrix} \right) . \end{aligned}$$
(6.1b)

For \(\mu \) near 0 it is convenient instead of (6.1a) to use a version of (3.16a):

$$\begin{aligned} \xi _p^{\pm }(k,x)= & {} \frac{\text {i}^{{\mp } \varepsilon _k \mu }}{2 N_p^{\pm }} \left( \varepsilon _k \mathcal {J}_{\varepsilon _k \lambda , \mu + \frac{1}{2}}(2|k|x) \begin{bmatrix} 1 \\ -z \end{bmatrix} \right. \nonumber \\&\quad \left. + \frac{\mathcal {J}_{\varepsilon _k \lambda , \mu - \frac{1}{2}} (2|k|x) +\varepsilon _k\lambda \mathcal {J}_{\varepsilon _k \lambda , \mu + \frac{1}{2}} (2|k|x)}{\mu } \begin{bmatrix} z \\ 1 \end{bmatrix} \right) . \end{aligned}$$
(6.2)

The leading terms of \(\xi _p^{\pm }\) and \(\zeta _p^\pm \) for large kx are

$$\begin{aligned}&\xi _p^{\pm }(k,x) \sim \frac{\text {e}^{- \frac{\varepsilon _k \pi \lambda }{2}}}{2} \left( \text {e}^{{\mp } \text {i}k x} (2 |k|x )^{{\mp } \text {i}\lambda } \begin{bmatrix} 1 \\ {\mp } \text {i}\end{bmatrix} + (S_p \text {e}^{- \text {i}\varepsilon _k \pi \mu })^{\pm 1} \text {e}^{\pm \text {i}k x} (2 |k| x)^{\pm \text {i}\lambda } \begin{bmatrix} 1 \\ \pm \text {i}\end{bmatrix} \right) , \end{aligned}$$
(6.3a)
$$\begin{aligned}&\zeta _p^{\pm }(k,x) \sim {\mp }\text {i}\, \text {e}^{\frac{\varepsilon _k \pi \lambda }{2}} \text {e}^{\pm \text {i}k x} (2 |k|x )^{\pm \text {i}\lambda } \begin{bmatrix} 1 \\ \pm \text {i}\end{bmatrix}. \end{aligned}$$
(6.3b)

Because of the long-range nature of the perturbation and of the presence of spin degrees of freedom, it is not obvious what should be chosen as the definition of the outgoing and incoming waves. Let us call \(\text {i}\zeta ^+(k,x)\) the outgoing wave and \(-\text {i}\zeta ^-(k,x)\) the incoming wave. Then the ratio of the outgoing wave and the incoming wave in \(\xi ^+(k,x)\) is \(\text {e}^{- \text {i}\varepsilon _k \mu } S_p\) and can be called the (full) scattering amplitude at energy k.

Proposition 30

Let \(p \in \mathcal {M}_{- \frac{1}{2}} \backslash (\mathcal {E}^+ \cup \mathcal {E}^-)\), \(k \in \mathbb {R}^\times \), \(s > |\text {Im}(\lambda )| + \frac{1}{2}\). Then the spectral density

$$\begin{aligned} \Pi _p(k):=(2 \pi \text {i})^{-1} \left( G^{\bowtie }_{p}(k + \text {i}0) - G^{\bowtie }_{p}(k - \text {i}0) \right) \end{aligned}$$
(6.4)

is well defined as a compact operator \( L^2_s(\mathbb {R}_+,\mathbb {C}^2) \rightarrow L^2_{-s}(\mathbb {R}_+,\mathbb {C}^2)\) and has the integral kernel

$$\begin{aligned} \Pi _{p}(k;x,y ) = \frac{\text {e}^{\varepsilon _k \pi \lambda }}{ \pi } \xi _{p}^{+}(k,x) \xi _{p}^{-}(k,y)^\text {T}= \frac{\text {e}^{\varepsilon _k \pi \lambda }}{ \pi } \xi _{p}^{-}(k,x) \xi _{p}^{+}(k,y)^\text {T}. \end{aligned}$$
(6.5)

As \(k \rightarrow 0\), it admits the expansion

$$\begin{aligned} \Pi _p(k) = \text {e}^{\varepsilon _k \pi \lambda } |k|^{2 \mu } \Pi _p^0 + O(|k|^{2 \mu +1}), \end{aligned}$$
(6.6)

where the remainder is estimated in the \(B(L^2_s,L^2_{-s})\) norm and \(\Pi _p^0\) has the integral kernel

$$\begin{aligned} \Pi _p^0(x,y) = \frac{ (4xy)^{ \mu }}{\pi \, \Gamma (2 \mu +1 )^2 N_p^+ N_p^-} \begin{bmatrix} z^2 &{} z \\ z &{} 1 \end{bmatrix}. \end{aligned}$$
(6.7)

Proof

The first statement follows from Proposition 27. By (3.27), it is sufficient to prove (6.5) for \(x < y \). Plugging (3.21a) into (3.26), we find

$$\begin{aligned}&G_p^{\bowtie }(k + \text {i}0;x,y) - G_p^{\bowtie }(k- \text {i}0;x,y) \nonumber \\&\quad = \xi _p^-(k,x) \left( \zeta _p^-(k,y) - \text {e}^{-\text {i}\varepsilon _k \pi \mu } S_p \zeta _p^+(k,y) \right) ^\text {T}. \end{aligned}$$
(6.8)

Plugging in (3.21b) we obtain (6.5).

The last part of the statement follows from asymptotics of \(\xi \) functions for small arguments and the dominated convergence theorem. \(\square \)

We refer to “Appendix C” for definitions used in the lemma below. Note also the identity \(\xi _p^\pm (k,x)=\xi _p^\pm (\varepsilon _k,|k|x)\), which allows us to restrict our attention to \(\xi _p^\pm (\varepsilon _k,x)\). The following fact follows immediately from Lemma 73 and (6.2).

Lemma 31

\( \xi _p^{\pm }(\varepsilon _k, x)\), \(p \notin \mathcal {E}^\pm \), is a tempered distribution in \(x \in \mathbb {R}_+\), in the sense explained in “Appendix C”. Its Mellin transform is

$$\begin{aligned} \Xi _p^\pm (\varepsilon _k, s)&:= \int _0^{\infty } \text {e}^{ - 0 x} x^{- \frac{1}{2} - \text {i}s} \xi _p^{\pm }(\varepsilon _k,x) \text {d}x \nonumber \\&= \text {i}^{{\mp } \varepsilon _k \mu - \frac{3}{2} - \mu + \text {i}s} 2^{\mu -1} \Gamma \left( \frac{1}{2} + \mu - \text {i}s \right) \frac{1}{N_p^{\pm }\mu } \nonumber \\&\quad \times \, \Bigg ( 2 \varepsilon _k \omega \Big (\frac{1}{2} + \mu - \text {i}s\Big ) \, {}_{2}\mathbf {F}_1 \left( 1 + \mu + \text {i}\varepsilon _k \lambda , { \scriptstyle \frac{3}{2}}\right. \nonumber \\&\quad \left. + \mu - \text {i}s ; 2 \mu + 2 ; 2 + \text {i}0 \right) \begin{bmatrix} - z \\ 1 \end{bmatrix} \nonumber \\&\quad +\, \text {i}\, {}_2 \mathbf {F}_1 \left( \mu + \text {i}\varepsilon _k \lambda , {\scriptstyle \frac{1}{2}} + \mu - \text {i}s ; 2 \mu ; 2 + \text {i}0 \right) \begin{bmatrix} z \\ 1 \end{bmatrix} \Bigg ), \end{aligned}$$
(6.9)

is analytic in s and bounded by \(c_p^{\pm } (1+s^2)^{\frac{1}{2} |\text {Im}(\lambda )|}\) locally uniformly in p.

We define \(\mathcal {U}_p^{\pm , \text {pre}}\), \(p \in \mathcal {M}\backslash \mathcal {E}^{\pm }\), as the integral operator \(C_\mathrm {c}^{\infty }(\mathbb {R}_+, \mathbb {C}^2) \rightarrow C^{\infty }(\mathbb {R}) \) with the kernel

$$\begin{aligned} \mathcal {U}^\pm _p(k,x) = \frac{\text {e}^{\frac{1}{2} \varepsilon _k \pi \lambda }}{ \sqrt{\pi }} \xi _p^{\pm }(k,x)^\text {T}. \end{aligned}$$
(6.10)

By construction, the kernel of the spectral density operator factors as

$$\begin{aligned} \Pi _p(k ; x , y) = \mathcal {U}_p^+(k,x)^{\text {T}} \mathcal {U}_p^-(k,y) = \mathcal {U}_p^-(k,x)^{\text {T}} \mathcal {U}_p^+(k,y). \end{aligned}$$
(6.11)

We note also the relations

$$\begin{aligned} \mathcal {U}_p^+(k,x) = \text {e}^{- \text {i}\varepsilon _k \pi \mu } S_{p} \, \mathcal {U}_p^-(k,x) , \qquad \overline{\mathcal {U}_{{\overline{p}}}^{\pm }(k,x)} = \mathcal {U}_{p}^{{\mp }}(k,x) \end{aligned}$$
(6.12)

and the intertwining property

$$\begin{aligned} (\mathcal {U}_p^{\pm , \text {pre}} D_p f)(k) = k (\mathcal {U}_p^{\pm , \text {pre}} f)(k), \qquad f \in C_\mathrm {c}^\infty (\mathbb {R}_+, \mathbb {C}^2). \end{aligned}$$
(6.13)

Recall from Sect. 1.1 that J is the inversion and A is the generator of dilations, and K is the multiplication operator on \(L^2(\mathbb {R})\) by the variable \(k\in \mathbb {R}\).

Below we will consider level sets \(\{ \lambda = \lambda _0 \} \subset \mathcal {M}_\frac{1}{2}\). Recall from the discussion around equation (2.9) that it is a submanifold for \(\lambda _0 \ne 0\), but for \(\lambda _0 =0\) it is the union of three submanifolds singular along the intersection. We will say that a function on the locus \(\{ \lambda =0 \}\) is holomorphic if its restriction to each of the three components is holomorphic.

Proposition 32

\(\mathcal {U}_p^{\pm , \text {pre}}\) are densely defined closable operators \(L^2(\mathbb {R}_+, \mathbb {C}^2) \rightarrow L^2(\mathbb {R})\) with the closures given by

$$\begin{aligned} \mathcal {U}_p^{\pm } f(k) = \frac{\text {e}^{\frac{1}{2} \varepsilon _k \pi \lambda }}{\sqrt{\pi }} \Xi _p^{\pm \text {T}}(\varepsilon _k, A) J f (|k|), \qquad k \in \mathbb {R}. \end{aligned}$$
(6.14)

Hence, \(\mathcal {U}^\pm _p (1+A^2)^{- \frac{1}{2} |\text {Im}(\lambda )|}\) is bounded. In particular \(\mathcal {U}^\pm _p\) are bounded if \(\lambda \in \mathbb {R}\). If \(\lambda _0 \in \mathbb {R}\), they form a holomorphic family of operators on the level set \(\{ \lambda = \lambda _0 \} \backslash \mathcal {E}^{\pm }\).

Furthermore, \(\mathcal {U}_p^{\pm *} = \mathcal {U}_{{\overline{p}}}^{{\mp } \text {T}}\).

Proof

The first part follows from Lemma 31 and discussion in “Appendix C”. Now fix \(\lambda _0 \in \mathbb {R}\) and consider p in a component S of the level set \(\{ \lambda = \lambda _0 \} \backslash \mathcal {E}^{\pm }\). If \(f \in C_c^{\infty }(\mathbb {R})\), \(g \in C_c^{\infty }(\mathbb {R}_+, \mathbb {C}^2)\), then \(( f | \mathcal {U}_{p}^{\pm } g )\) is a holomorphic function of \(p \in S\). Since \(C_c^{\infty }\) spaces are dense in \(L^2\) and \(\mathcal {U}_p^{\pm }\) are bounded locally uniformly in p, \(\mathcal {U}_p^{\pm }\) is a holomorphic operator-valued function. The last claim follows from the formula (6.12). \(\square \)

In a sense, operators \(\mathcal {U}^\pm _p\) diagonalize \(D_p\) for \(p \in \mathcal {M}_{- \frac{1}{2}} \backslash \mathcal {E}^\pm \). If \(p = {\overline{p}}\), then \(D_p\) are self-adjoint and \(\mathcal {U}_p^\pm \) are unitary. If we assume only that \(\lambda \) is real, then \(\mathcal {U}_p^\pm \) are still bounded with bounded inverses, so they are almost as good as in the self-adjoint case. This will be made precise below.

Proposition 33

If \(p = {\overline{p}}\), then for any bounded interval \([a,b] \subset {\mathbb {R}}\) and \(f,g \in C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\)

$$\begin{aligned} ( g|\mathbbm {1}_{[a,b]}(D_p) f ) = \int _a^b \int _{0}^{\infty } \int _{0}^{\infty } g(x)^* \Pi _{p}(k;x,y ) f(y ) \text {d}y \text {d}x \text {d}k. \end{aligned}$$
(6.15)

Besides, \(\mathcal {U}_{p}^\pm \) is a unitary operator and

$$\begin{aligned} D_{p} = \mathcal {U}_{p}^{\pm *} K \mathcal {U}_{p}^{\pm }. \end{aligned}$$
(6.16)

Proof

Since the point spectrum of \(D_{p}\) is trivial for \(p={\overline{p}}\), Stone’s formula gives

$$\begin{aligned} ( g|\mathbbm {1}_{[a,b]}(D_p) f )= & {} \lim _{\epsilon \downarrow 0} \frac{1}{2 \pi \text {i}} \int _{[a,b] \times \mathbb {R}_+^2} g(x)^* (G_{p}^{\bowtie }(k + \text {i}\epsilon ;x,y ) \nonumber \\&\quad -\, G_{p}^{\bowtie }(k - \text {i}\epsilon ;x,y )) f(y ) \text {d}y \text {d}x \text {d}k. \end{aligned}$$
(6.17)

It follows from the asymptotics of functions \(\xi _p^{\pm }\) and \(\zeta _p\) that on \([a,b] \times \text {supp}(f) \times \text {supp}(g)\) we have \(|G_{p}^{\bowtie }( k \pm \text {i}\epsilon ;x,y )| \le c |k|^{ \text {Re}(\mu )-|\text {Re}(\mu )|}\) with c independent of k. This function is integrable, because \( \text {Re}(\mu )-|\text {Re}(\mu )| > -1\). Therefore, by the dominated convergence theorem, the limit \(\epsilon \downarrow 0\) may be taken under the integral. This proves (6.15).

Let us prove the unitarity of \(\mathcal {U}_p^\pm \). Let \(f \in C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\) and let [ab] be a bounded interval. Then

$$\begin{aligned} \int _a^b |\mathcal {U}_{p}^{\pm } f(k)|^2 \text {d}k&= \int _0^{\infty } \int _a^b \int _0^{\infty } f(x)^*\Pi _{p}(k;x,y ) f(y ) \text {d}y \text {d}k \text {d}x \nonumber \\&= \int _0^{\infty } f(x)^* (\mathbbm {1}_{[a,b]}(D_p)f)(x) \text {d}x = ( f| \mathbbm {1}_{[a,b]}(D_p) f ), \end{aligned}$$
(6.18)

where in the first step we used the definition of \(\mathcal {U}_{p}^\pm \), conjugation formula (6.12) and the factorization (6.11). The order of integrals is immaterial, because the integrand is compactly supported and its only possible singularity (at \(k=0\), if \(0 \in [a,b]\)) is integrable. In the second step we used Proposition 33. Taking the limit \(b \rightarrow \infty \), \(a \rightarrow - \infty \) we find

$$\begin{aligned} \int _{- \infty }^{\infty } |\mathcal {U}_{p}^{\pm } f(k)|^2 \text {d}k = ( f| f ). \end{aligned}$$
(6.19)

Hence, \(\mathcal {U}_{p}^{\pm }\) is an isometry. Equation (6.18) implies that

$$\begin{aligned} \mathbbm {1}_{[a,b]}(D_p) = \mathcal {U}_p^{\pm *} \mathbbm {1}_{[a,b]}(K) \mathcal {U}_p^{\pm }. \end{aligned}$$
(6.20)

It remains to show that \(\mathcal {U}_p^{\pm }\mathcal {U}_p^{\pm *}=1\). The proof of this fact follows closely the proof of (3.37) of Theorem 3.16 in [13]. \(\square \)

Proposition 34

If \(p \in \mathcal {M}_{- \frac{1}{2}} \backslash (\mathcal {E}^+ \cup \mathcal {E}^-)\) is such that \(\lambda \in \mathbb {R}\), then \((\mathcal {U}_p^\pm )^{-1}=\mathcal {U}^{{\mp }\text {T}}\) and

$$\begin{aligned} D_p = \mathcal {U}_p^{\pm -1} K \mathcal {U}_p^{\pm }. \end{aligned}$$
(6.21a)

In particular \(D_p\) is similar to a self-adjoint operator.

Proof

We fix \(\lambda _0 \in \mathbb {R}\). Then \(\mathcal {U}_p^{\pm } \mathcal {U}_p^{{\mp } \text {T}}-1\) and \(\mathcal {U}_p^{\pm \text {T}} \mathcal {U}_p^{{\mp }}-1\) form holomorphic families of bounded operators on (one-dimensional) \(\{ \lambda = \lambda _0 \} \backslash (\mathcal {E}^+ \cup \mathcal {E}^-)\). They vanish on the set of real points, which has an accumulation point in each component of the domain. Thus, they vanish everywhere.

Now take \(k \in \mathbb {C}\backslash \mathbb {R}\). Arguing as in the previous paragraph, we obtain

$$\begin{aligned} \mathcal {U}_p^{\pm -1} (K-k)^{-1} \mathcal {U}_p^\pm = (D_p - k)^{-1}, \end{aligned}$$
(6.22)

from which (6.21a) follows immediately. \(\square \)

Question 2

If \(\lambda \in \mathbb {R}\), then \(D_p\) is similar to a self-adjoint operator. Hence, it enjoys a very good functional calculus–for any bounded Borel function f the operator \(f(D_p)\) is well defined and bounded.

If \(\lambda \notin \mathbb {R}\) this is probably no longer true, because the diagonalizing operators \(\mathcal {U}_p^\pm \) are unbounded. However, they are unbounded in a controlled manner: they are continuous on the domain of some power of the dilation operator. One may hope that this is sufficient to allow for a rich functional calculus for Dirac–Coulomb Hamiltonians. We pose an open problem: for a given \(\text {Im}(\lambda )\), characterize functions that allow for a functional calculus for \(D_p\). In particular, one could ask when \(\text {i}D_p\) generates a \(C^0\) semigroup of bounded operators.

Numerical Range and Dissipative Properties

In this section we give a complete analysis of the numerical range of various realizations of 1d Dirac–Coulomb Hamiltonians studied in this paper.

Proposition 35

One of the following mutually exclusive statements is true:

  1. 1.

    \(\omega \) and \(\lambda \) are real. Then \(\text {Num}(D_{\omega , \lambda }^\text {pre}) = \mathbb {R}\).

  2. 2.

    \(|\text {Im}(\omega )| < |\text {Im}(\lambda )|\). Then \(\text {Num}(D_{\omega , \lambda }^\text {pre}) = \mathbb {C}_{- \text {sgn}(\text {Im}(\lambda ))}\).

  3. 3.

    \(|\text {Im}(\omega )| = |\text {Im}(\lambda )| \ne 0\). Then \(\text {Num}(D_{\omega , \lambda }^\text {pre}) = \mathbb {C}_{- \text {sgn}(\text {Im}(\lambda ))} \cup \{ 0 \}\).

  4. 4.

    \(|\text {Im}(\omega )| > |\text {Im}(\lambda )|\). Then \(\text {Num}(D_{\omega , \lambda }^\text {pre}) = \mathbb {C}\).

The same is true with \(D_{\omega , \lambda }^\text {pre}\) replaced by \(D_{\omega , \lambda }^{\min }\) throughout.

Proof

Integrating by parts we find that for \(f = \begin{bmatrix} f_1 \\ f_2 \end{bmatrix} \in C_\mathrm {c}^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\) we have

$$\begin{aligned} \text {Im}( f| D_{\omega , \lambda } f )= & {} - \text {Im}(\lambda + \omega ) \int _0^{\infty } \frac{|f_1(x)|^2}{x} \text {d}x \nonumber \\&\quad -\, \text {Im}(\lambda - \omega ) \int _0^{\infty } \frac{|f_2(x)|^2}{x} \text {d}x . \end{aligned}$$
(7.1)

In the four cases listed in the proposition we have: both terms are zero in Case 1., both terms are nonzero (except for \(f=0\)) and have the same sign as \(- \text {Im}(\lambda )\) in Case 2., one term is zero and the other has the same sign as \(- \text {Im}(\lambda )\) in Case 3. and the two terms have opposite signs in the last case. Therefore, inclusions of numerical ranges in the specified sets are clear, except for the third case. Then in order for \(\text {Im}( f| D_{\omega , \lambda } f )\) to vanish, one of the two \(f_j\) has to be zero. It is easy to check that this implies \(( f| D_{\omega , \lambda } f ) =0\) (but not \(f =0\)).

We have to show that the obtained inclusions are saturated. The homogeneity of \(D_{\omega , \lambda }^\text {pre}\) implies that \(\text {Num}(D_{\omega , \lambda }^\text {pre})\) is a convex cone. Thus, to establish the result in Case 1. it is sufficient to show that both signs of \(( f | D_{\omega , \lambda } f )\) are possible. We choose a nonzero \(\varphi \in C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\) with \(\Vert \varphi \Vert _{H_0^1}=1\) and put \(f_{\pm , t}(x) = \begin{bmatrix} \varphi (x-t) \\ \pm \varphi '(x-t) \end{bmatrix}\) for \(t \ge 0\). Then \(\Vert f_{\pm , t} \Vert _{L^2}=1\) and

$$\begin{aligned} ( f_{\pm , t} | D_{\omega ,\lambda } f_{\pm , t} )= & {} \pm 2 \int _0^{\infty } |\varphi '(x)|^2 \text {d}x \nonumber \\&\quad -\, \int _0^{\infty } \frac{1}{x+t} \left( (\lambda +\omega ) |\varphi (x)|^2 + (\lambda - \omega ) |\varphi '(x)|^2 \right) \text {d}x . \end{aligned}$$
(7.2)

The first term is nonzero, has sign ± and does not depend on t, while the other converges to zero for \(t \rightarrow \infty \). Therefore, \(\pm ( f_{\pm , t} |D_{\omega , \lambda } f_{\pm , t} ) \ge c_{\pm } >0\) for large enough t.

Next we suppose that \(|\text {Im}(\omega )| \le |\text {Im}(\lambda )| \ne 0\). It is sufficient to show that \(\mathbb {C}_-\) is included in the numerical range for \(\text {Im}(\lambda )<0\). Arguing as below (7.2), we deduce that there exist constants \(t_0 > 0\) and \(c_{\pm } >0\) such that \(\pm \text {Re}\, ( f_{\pm , t} | D_{\omega ,\lambda } f_{\pm , t} ) \ge c_{\pm }\) for \(t \ge t_0\). Let \(\delta = \text {Im}\, (f_{ \pm , t_0} |D_{\omega ,\lambda } f_{ \pm , t_0} )\). Then \(\delta >0 \). The function \(t \mapsto \text {Im}\, ( f_{\pm , t} | D_{\omega ,\lambda } f_{\pm , t} )\) is continuous and converges to zero for \(t \rightarrow \infty \), so for every \(\epsilon \in ]0, \delta ]\) there exists \(t \ge t_0\) such that \(\text {Im}\, ( f_{\pm , t} | D_{\omega ,\lambda } f_{\pm , t} ) = \epsilon \). By convexity of numerical ranges this implies \([-c_- + \text {i}\epsilon , c_+ + \text {i}\epsilon ] \subset \text {Num}(D_{\omega ,\lambda })\). Homogeneity implies that for every \(s>0\) we have \(\left[ -\frac{c_- s}{\epsilon } + \text {i}s, \frac{c_+ s}{\epsilon } + \text {i}s \right] \subset \text {Num}(D_{\omega , \lambda })\). Every k with \(\text {Im}(k)=s\) is in this interval for small enough \(\epsilon \).

Similar argument shows that in Case 4. there exist \(c_{\pm } >0\) and \(\delta >0\) such that for every \(\epsilon \in ] 0 , \delta ]\) there exist \(g_{\pm ,\epsilon } \in C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\) with \(\Vert g_{\pm ,\epsilon } \Vert _{L^2}=1\), \(\pm \text {Re}( g_{\pm ,\epsilon }|D_{\omega , \lambda } g_{\pm ,\epsilon } ) \ge c_{\pm }\) and \(\left| \text {Im}( g_{\pm ,\epsilon }| D_{\omega , \lambda } g_{\pm ,\epsilon } )\right| \le \epsilon \). On the other hand for nonzero \(f \in C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\) with \(f_1=0\) or \(f_2=0\), we have that \(( f| D_{\omega , \lambda } f )\) is proportional to \(\omega -\lambda \) or \(- \omega - \lambda \), respectively, with a positive proportionality constant. Using homogeneity we can even construct functions f with the proportionality constant equal to 1 and \(\Vert f \Vert =1\). Next we observe that if \(\epsilon \) is taken to be sufficiently small, the convex hull of \(( g_{+, \epsilon } | D_{\omega , \lambda } g_{+, \epsilon } )\), \(( g_{-, \epsilon } | D_{\omega , \lambda } g_{-, \epsilon } )\), \(\omega - \lambda \) and \(- \omega - \lambda \) contains zero in its interior. Therefore, the smallest convex cone containing it is the whole \(\mathbb {C}\).

To prove the last statement, first note that \(\text {Num}(D_{\omega , \lambda }^{\min })\) is contained in the closure of \(\text {Num}(D_{\omega , \lambda }^\text {pre})\). Therefore, in Cases 1. and 4. there is nothing to prove. We consider Case 2. We have to show that if \(g \in \text {Dom}(D_{\omega , \lambda }^{\min })\) is such that \(\text {Im}( g| D_{\omega , \lambda } g ) =0\), then \(g=0\). We choose \(\epsilon >0\) and \(f \in C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2)\) such that \(\Vert f - g \Vert _{\text {Dom}(D_{\omega , \lambda }^{\min })} < \epsilon \). Then

$$\begin{aligned} \text {Im}( f| D_{\omega , \lambda } f )= & {} \text {Im}\left( ( g | D_{\omega , \lambda }^{\min } (f-g) ) \right. \nonumber \\&\quad \left. +\, ( f-g | D_{\omega , \lambda }^{\min } g ) + (f-g | D_{\omega , \lambda }^{\min } (f-g) ) \right) , \end{aligned}$$
(7.3)

so \(|\text {Im}( f| D_{\omega , \lambda } f )| \le 2 \epsilon \Vert g \Vert _{\text {Dom}(D_{\omega , \lambda }^{\min })} + \epsilon ^2\). On the other hand for any \(t >0\), we have

$$\begin{aligned} |\text {Im}( f| D_{\omega , \lambda } f )|&\ge \frac{|\text {Im}(\lambda +\omega )|}{t} \int _0^t |f_1(x)|^2 \text {d}x + \frac{|\text {Im}(\lambda -\omega )|}{t} \int _0^t |f_2(x)|^2 \text {d}x s\nonumber \\&\ge \frac{|\text {Im}(\lambda +\omega )|}{t} \left( \int _0^t |g_1(x)|^2 \text {d}x - 2 \epsilon \Vert g \Vert ^2 \right) \nonumber \\&\quad + \frac{|\text {Im}(\lambda -\omega )|}{t} \left( \int _0^t |g_2(x)|^2 \text {d}x - 2 \epsilon \Vert g \Vert ^2 \right) . \end{aligned}$$
(7.4)

Comparing the two derived inequalities and taking \(\epsilon \rightarrow 0\) we find that

$$\begin{aligned} \int _0^t |g_1(x)|^2 \text {d}x = \int _0^t |g_2(x)|^2 \text {d}x = 0. \end{aligned}$$
(7.5)

Since t was arbitrary, \(g=0\). Case 3. may be handled analogously. \(\square \)

It is convenient to describe the numerical ranges of operators \(D_p\) in terms of [a : b] . It can be related to parameters \(\omega , \lambda , \mu \) by recalling that \([a:b] = [- \mu : \omega + \lambda ]\) if \(\omega +\lambda \ne 0\) and \([a:b] = [\omega - \lambda : - \mu ]\) if \(\omega - \lambda \ne 0\). No such expression exists on the zero fiber. We will also choose a representative \((a,b) \in [a:b]\). We note that the condition \(\text {Im}({\overline{b}} a) =0\) is equivalent to the existence of a real representative (ab), which is also equivalent to the statement that [a : b] belongs to the real projective line \(\mathbb {R}\mathbb {P}^1\). If \([a:b] \notin \mathbb {R}\mathbb {P}^1\), then \(\text {sgn}(\text {Im}({\overline{b}} a)) = \text {sgn}\left( \text {Im}\left( \frac{a}{b} \right) \right) \).

Proposition 36

The numerical range of \(D_p\) may be characterized as follows.

  1. 1.

    If \(\omega , \lambda \in \mathbb {R}\) and \([a:b] \notin \mathbb {R}\mathbb {P}^1\), then \(\text {Num}(D_p) = \mathbb {R}\cup \mathbb {C}_{- \text {Im}\left( {\overline{b}} a \right) }\).

  2. 2.

    If \(\text {Re}(\mu )=0\) and \(\text {Im}({\overline{b}} a) \text {Im}(\lambda )<0\), then \(\text {Num}(D_p) = \mathbb {C}\).

  3. 3.

    If \(\text {Re}(\mu )<0\) and \([a:b] \notin \mathbb {R}\mathbb {P}^1\), then \(\text {Num}(D_p) = \mathbb {C}\).

  4. 4.

    In every other case \(\text {Num}(D_p) = \text {Num}(D_{\omega , \lambda }^{\min })\).

Proof

If \(p = {\overline{p}}\), then \(D_p\) is self-adjoint, so \(\text {Num}(D_p) \subset \mathbb {R}= \text {Num}(D_{\omega , \lambda }^{\min }) \subset \text {Num}(D_p)\). If \(|\text {Im}(\omega )| > |\text {Im}(\lambda )|\), then \(\mathbb {C}= \text {Num}(D_{\omega , \lambda }^{\min }) \subset \text {Num}(D_{p})\).

Let \(\eta (x) = x^{\mu } \begin{bmatrix} a \\ b \end{bmatrix}\) and consider \(f = \begin{bmatrix} f_1 \\ f_2 \end{bmatrix} \in C_c^{\infty }(\mathbb {R}_+,\mathbb {C}^2) + \mathrm {span} \{ \chi \eta \}\). Then

$$\begin{aligned} \text {Im}( f | D_{p} f )= & {} \text {Im}\int _{0}^{\infty } \left[ \frac{\text {d}}{\text {d}x} \left( \overline{f_2(x)} f_1(x) \right) \right. \nonumber \\&\quad \left. -\, \frac{(\lambda + \omega ) |f_1(x)|^2 + (\lambda - \omega ) |f_2(x)|^2}{x} \right] \text {d}x. \end{aligned}$$
(7.6)

By construction, there exist \(x_0 >0\) and \(c \in \mathbb {C}\) such that for \(x < x_0\) we have \(f(x) = c \, \eta (x)\), and hence, \(\overline{f_2(x)} f_1(x) = \text {Im}({\overline{b}} a) x^{2 \text {Re}(\mu )}\). If \(\text {Re}(\mu )>0\) or \(\text {Im}({\overline{b}} a)=0\) (which is equivalent to \([a:b] \in \mathbb {R}\mathbb {P}^1 \subset \mathbb {C}\mathbb {P}^1\)), then \(\overline{f_2(x)} f_1(x)\) vanishes for x sufficiently large and for \(x \rightarrow 0\). Therefore, \(\int _{0}^{\infty } \frac{\text {d}}{\text {d}x} \left( \overline{f_2(x)} f_1(x) \right) \text {d}x =0\) and the proof goes as for Proposition 35.

We consider the case \(\text {Re}(\mu ) = 0\) and \(\text {Im}({\overline{b}} a) \ne 0\). Then

$$\begin{aligned} \text {Im}( f | D_p f )= & {} - |c|^2 \text {Im}({\overline{b}} a) \nonumber \\&\quad -\, \text {Im}\int _{0}^{\infty } \frac{(\lambda + \omega ) |f_1(x)|^2 + (\lambda - \omega ) |f_2(x)|^2}{x} \text {d}x. \end{aligned}$$
(7.7)

If \(\omega , \lambda \in \mathbb {R}\), then \(\text {Im}( f | D_p f ) = - |c|^2 \text {Im}({\overline{b}} a)\) and we have \(\mathbb {R}= \text {Num}(D_{\omega , \lambda }^{\min }) \subset \text {Num}(D_p)\), so \(\text {Num}(D_p) = \{ k \in \mathbb {C}\, | \, \text {Im}({\overline{b}} a) \text {Im}(k) \le 0 \}\). In the case \(|\text {Im}(\omega )| \le |\text {Im}(\lambda )| \ne 0\) there are two possibilities. If \(\text {Im}({\overline{b}} a) \text {Im}(\lambda )>0\), then all terms in (7.7) have the same sign and one has \(\text {Num}(D_p) = \text {Num}(D_{\omega , \lambda }^{\min })\). Otherwise \(\text {Num}(D_p) = \mathbb {C}\). Indeed, consider \(f = \frac{\chi \eta }{\Vert \chi \eta \Vert }\) with shrinking support of \(\chi \ge 0\). A simple calculation shows that for these functions the integrand in (7.7) vanishes, while the first term grows without bound.

Next, we suppose that \(\text {Re}(\mu ) <0\), \(\text {Im}({\overline{b}} a) \ne 0\). Put \(f = \varphi \eta \) with \(\varphi \in C^{\infty }([0 ,\infty [, \mathbb {R})\) vanishing exponentially at infinity. Then \(f \in \text {Dom}(D_p)\) and \((D_p f)(x) = \varphi '(x) \begin{bmatrix} 0 &{} -1 \\ 1 &{} 0 \end{bmatrix} \eta (x)\). Thus,

$$\begin{aligned} ( f | D_p f ) = 2 \text {i}\, \text {Im}({\overline{b}} a) \int _0^{\infty } \varphi (x) \varphi '(x) x^{2 \text {Re}(\mu )} \text {d}x. \end{aligned}$$
(7.8)

If \(\varphi \ne 0\) vanishes at zero, the integral is positive, as can be seen by integrating by parts:

$$\begin{aligned} \int _0^{\infty } \varphi (x) \varphi '(x) x^{2 \text {Re}(\mu )} \text {d}x= & {} \frac{1}{2} \int _0^{\infty } \frac{\text {d}}{\text {d}x} (\varphi (x)^2) x^{2 \text {Re}(\mu )} \text {d}x \nonumber \\= & {} - \text {Re}(\mu ) \int _0^{\infty } \varphi (x)^2 x^{2 \text {Re}(\mu )-1} \text {d}x > 0. \end{aligned}$$
(7.9)

On the other hand, for \(\varphi (x) = \text {e}^{- \frac{x}{2}}\) the integral is negative:

$$\begin{aligned} \int _0^{\infty } \varphi (x) \varphi '(x) x^{2 \text {Re}(\mu )} \text {d}x = - \frac{\Gamma (2 \text {Re}(\mu )+1)}{2} <0. \end{aligned}$$
(7.10)

By Proposition 35 and the fact that \(\text {Num}(D_p)\) is a convex cone, we have \(\text {Num}(D_p) = \mathbb {C}\). \(\square \)

We adopt the convention saying that operators with the numerical range contained in the closed upper half-plane are called dissipative. Dissipative operators which are not properly contained in another dissipative operator are said to be maximally dissipative. This condition is equivalent to the inclusion of the spectrum in the closed upper half plane. Maximally dissipative operators may also be characterized as operators D such that \(\text {i}D\) is the generator of a semigroup of contractions.

Corollary 37

\(\pm D_p\) is a dissipative operator if and only if one of the following (mutually exclusive) statements holds:

  • \(\omega , \lambda \in \mathbb {R}\) and \({\mp }\text {Im}({\overline{b}} a) \ge 0\).

  • \(\pm \text {Im}(\lambda )<0\), \(|\text {Im}(\omega )| \le |\text {Im}(\lambda )|\) and \(\text {Re}(\mu ) >0\).

  • \(\pm \text {Im}(\lambda )<0\), \(|\text {Im}(\omega )| \le |\text {Im}(\lambda )|\), \(\text {Re}(\mu ) =0\) and \(\pm \text {Im}({\overline{b}} a) \le 0\).

  • \(\pm \text {Im}(\lambda )<0\), \(|\text {Im}(\omega )| \le |\text {Im}(\lambda )|\), \(\text {Re}(\mu )<0\) and \(\text {Im}({\overline{b}} a) =0\).

Furthermore, if these conditions are satisfied then \(\pm D_p\) is maximally dissipative.

Corollary 38

Let \(\omega , \lambda \) be such that \(\pm D_{\omega , \lambda }^{\min }\) is dissipative, i.e., \(|\text {Im}(\omega )| \!\le \! |\text {Im}(\lambda )|\), \(\pm \text {Im}(\lambda ) \le 0\). There exists \(p \in \mathcal {M}_{- \frac{1}{2}}\) such that \(D_{\omega , \lambda }^{\min } \subset D_p\) and \(\pm D_p\) is maximally dissipative. In particular \(\pm D_{\omega , \lambda }^{\min }\) admits a maximally dissipative extension which is homogeneous and contained in \(\pm D_{\omega , \lambda }^{\max }\).

Proof

We present the proof for the upper sign. The other part of the statement then follows by taking complex conjugates. If \(\omega , \lambda \in \mathbb {R}\), it is possible to choose p with \({\mp } \text {Im}({\overline{b}} a) \ge 0\). Now let \(\text {Im}(\lambda ) <0\), \(|\text {Im}(\omega )| \le |\text {Im}(\lambda )|\). If \(\omega ^2 - \lambda ^2 \notin ] - \infty , 0 ]\), we can choose \(\mu \) with \(\text {Re}(\mu ) >0\).

Next suppose that \(\omega ^2 - \lambda ^2 \le 0\). If the inequality is strict, then there exist two possible choices of \(\mu \) differing by a sign, so the condition \(\text {Im}({\overline{b}} a) \le 0\) is satisfied for at least one choice. If \(\omega ^2 - \lambda ^2 =0\), then either \(\omega + \lambda \) or \(\omega - \lambda \) vanishes. We may assume that it is not true that both vanish, because this is covered by the case \(\omega , \lambda \in \mathbb {R}\). Then \([a:b]=[0:1]\) or \([a:b]=[1:0]\). \(\square \)

Mixed Boundary Conditions

In this section we discuss operators \(D_{\omega ,\lambda }^f\) introduced around equation (4.8). Hence, \(\omega ,\lambda \) are restricted to the region \(| \text {Re}\sqrt{\omega ^2 - \lambda ^2}| < \frac{1}{2}\).

Proposition 39

\(D_{\omega ,\lambda }^f\) is closed, self-transposed and \(\sigma _{\text {ess}}(D_{\omega ,\lambda }^f) = \sigma _{\text {ess},0}(D_{\omega ,\lambda }^f) = \mathbb {R}\).

Proof

The self-transposedness follows from [11, Proposition 3.21]. The statement about the essential spectrum follows from Corollary 26. \(\square \)

Operators \(D_{\omega ,\lambda }^f\) can be organized in a holomorphic family as follows. Let

$$\begin{aligned} \mathcal {M}^{\text {mix}} = \left\{ (\omega ,\lambda ,[a:b]) \in \mathbb {C}^2 \times \mathbb {C}\mathbb {P}^1 \, | \, \text {Re}\sqrt{\omega ^2 - \lambda ^2}| < \frac{1}{2} \right\} . \end{aligned}$$
(8.1)

We define \(D^\text {mix}_{\omega ,\lambda ,[a:b]}\) to be \(D_{\omega ,\lambda }^{f_{\omega ,\lambda ,[a:b]}}\), where \(f_{\omega ,\lambda ,[a:b]}\) is a (unique up to a multiplicative constant) solution of \(D_{\omega ,\lambda } f_{\omega ,\lambda ,[a:b]} =0\) whose value at \(x=1\) belongs to the ray [a : b] in \(\mathbb {C}^2\).

Proposition 40

\(D_{\omega ,\lambda ,[a:b]}^\text {mix}\) form a holomorphic family of operators on \(\mathcal {M}^\text {mix}\). One has \(\overline{D_{\omega ,\lambda ,[a:b]}^\text {mix}} = D_{\overline{\omega },{\overline{\lambda }},[\overline{a}:\overline{b}]}^\text {mix}\), so \(D_{\omega ,\lambda ,[a:b]}^\text {mix}\) is self-adjoint if and only if \(\omega ,\lambda \in \mathbb {R}\), \([a:b] \in \mathbb {R}\mathbb {P}^1\).

Proof

Only the holomorphy of \(D_{\omega ,\lambda ,[a:b]}^\text {mix}\) requires some justification. Define

$$\begin{aligned} T_{\omega ,\lambda ,[a:b]} : H_0^1(\mathbb {R}_+,\mathbb {C}^2) \oplus \mathbb {C}\ni (g,t) \mapsto g + t \chi f_{\omega ,\lambda ,[a:b]}, \end{aligned}$$
(8.2)

where \(\chi \in C_c^\infty ([\mathbb {R}_+,\infty [)\) is equal to 1 near 0. It is easy to check that \(T_{\omega ,\lambda ,[a:b]}\) form a holomorphic family of bounded injective operators with \(\text {Ran}(T_{\omega ,\lambda ,[a:b]}) = \text {Dom}(D_{\omega ,\lambda ,[a:b]}^{\text {mix}})\) such that \(D_{\omega ,\lambda ,[a:b]}^{\text {mix}} T_{\omega ,\lambda ,[a:b]}\) form a holomorphic family of bounded operators. \(\square \)

Next we describe the point spectra of nonhomogeneous operators \(D_{\omega ,\lambda }^f\). For this purpose it is not very convenient to use the parametrization by points of \(\mathcal {M}^\text {mix}\).

Below we treat the logarithm, denoted \(\mathrm {Ln}\), as a set-valued function, more precisely,

$$\begin{aligned} \mathrm {Ln}(z):=\{u\ |\ z=\text {e}^u\}. \end{aligned}$$
(8.3)

Proposition 41

Consider the point spectrum of \(D_{\omega ,\lambda }^f\) for various \(\omega ,\lambda , f\). All eigenvalues are non-degenerate and zero is never an eigenvalue. For \(k \ne 0\), we split the discussion into several cases. We say that a pair \((k,\pm )\) is admissible if either \(k \in \mathbb {R}^\times \), \(|\text {Im}(\lambda )| > \frac{1}{2}\) and \(\pm = \text {sgn}(\text {Im}(\lambda ))\) or \(k \in \mathbb {C}{\setminus } \mathbb {R}\) and \(\pm = \text {sgn}(\text {Im}(k))\).

  1. 1.

    Case \(\mu \ne 0\). We select select a square root \(\mu = \sqrt{\omega ^2 - \lambda ^2}\), or equivalently, we fix \(p \in \mathcal {M}_{-\frac{1}{2}}\) lying over \(\omega ,\lambda \). All nonhomogeneous realizations of \(D_{\omega ,\lambda }\) correspond to

    $$\begin{aligned} f(x) = \begin{bmatrix} \omega - \lambda \\ - \mu \end{bmatrix} x^\mu + \kappa \begin{bmatrix} \omega - \lambda \\ \mu \end{bmatrix} x^{- \mu }\end{aligned}$$
    (8.4)

    with \(\kappa \in \mathbb {C}^\times \). Let

    $$\begin{aligned} c_{p,\pm } = \frac{\omega }{\lambda {\mp } \text {i}\mu } \frac{\Gamma (2 \mu +1)}{\Gamma (-2 \mu +1)} \frac{\Gamma (1 - \mu {\mp } \text {i}\lambda )}{\Gamma (1 + \mu {\mp } \text {i}\lambda )} \end{aligned}$$
    (8.5)

    Away from \(\mu =0\), \(c_{p,\pm }\) is a holomorphic function of \(\omega ,\lambda , \mu \) valued in \(\mathbb {C}\cup \{ \infty \}\). k is an eigenvalue if and only if \(\kappa ({\mp } 2 \text {i}k)^{2 \mu } = c_{p, \pm }\) and \((k, \pm )\) is admissible. \(D_{\omega ,\lambda }^f\) has no eigenvalues in \(\mathbb {C}_\pm \) if \(c_{p, \pm } \in \{ 0 ,\infty \}\). Away from these loci, eigenvalues in \(\mathbb {C}_\pm \) vary continuously with parameters, possibly (dis)appearing on the real axis. They form a discrete subset of a half-line if \(\mu \in \text {i}\mathbb {R}\), of a circle if \(\mu \in \mathbb {R}\) and of a logarithmic spiral otherwise. If \(\mu \not \in \text {i}\mathbb {R}\), the set of eigenvalues is finite. More precisely, it is given by the union of the following two sets:

    $$\begin{aligned} \left\{ k=\pm \frac{\text {i}}{2}\text {e}^w\ |\ w\in \frac{1}{2\mu }\mathrm {Ln}(c_{p,\pm }),\quad -\frac{\pi }{2}<\mathrm {Im} (w)< \frac{\pi }{2} \right\} ,&\quad |\text {Im}(\lambda )|\le \frac{1}{2}, \end{aligned}$$
    (8.6)
    $$\begin{aligned} \left\{ k=\pm \frac{\text {i}}{2}\text {e}^w\ |\ w\in \frac{1}{2\mu }\mathrm {Ln}(c_{p,\pm }),\quad -\frac{\pi }{2}\le \mathrm {Im}(w)\le \frac{\pi }{2} \right\} ,&\quad |\text {Im}(\lambda )|>\frac{1}{2}. \end{aligned}$$
    (8.7)
  2. 2.

    Case \(\mu =0\), \((\omega ,\lambda )\ne (0,0)\). All nonhomogeneous realizations of \(D_{\omega ,\lambda }\) are parametrized by \(\nu \in \mathbb {C}\) and

    $$\begin{aligned} f(x) = \begin{bmatrix} 1 \\ 0 \end{bmatrix} -2 \lambda ( \ln (\text {e}^{2\gamma } x) + \nu ) \begin{bmatrix} 0 \\ 1 \end{bmatrix}&\quad \text {for}\quad \omega =\lambda \ne 0, \end{aligned}$$
    (8.8)
    $$\begin{aligned} f(x) = \begin{bmatrix} 0 \\ 1 \end{bmatrix} + 2 \lambda ( \ln (\text {e}^{2 \gamma } x) + \nu ) \begin{bmatrix} 1 \\ 0 \end{bmatrix}&\quad \text {for}\quad \omega = - \lambda \ne 0.\end{aligned}$$
    (8.9)

    In both cases k is an eigenvalue if and only if \(\ln ({\mp } 2 \text {i}k) + \psi (1 {\mp } \text {i}\lambda ) {\mp } \frac{\text {i}}{2 \lambda } = \nu \) and \((k,\pm )\) is admissible. There is at most one eigenvalue in \(\mathbb {C}_+\) and at most one eigenvalue in \(\mathbb {C}_-\). The eigenvalue in \(\mathbb {C}_\pm \) exists if and only if \(\pm \text {i}\lambda \not \in \mathbb {N}\) and \(\text {Re}\left( \exp \left( \nu - \psi (1 {\mp } \text {i}\lambda ) {\mp } \frac{\text {i}}{2 \lambda } \right) \right) >0 \).

  3. 3.

    Case \(\omega = \lambda =0\), \(f(x) = \begin{bmatrix} 1 \\ \kappa \end{bmatrix}\). k is an eigenvalue if and only if \(k \not \in \mathbb {R}\) and \(\kappa = \text {i}\, \text {sgn}(\text {Im}(k))\).

Proof

An eigenvector of \(D_{\omega ,\lambda }\) square integrable away from the origin is necessarily of the form \(\zeta _p^\pm (k,\cdot )\) with an admissible \((k , \pm )\). It belongs to the domain of \(D_{\omega ,\lambda }^f\) if its asymptotic form for \(x \rightarrow 0\), obtained from (D.24), is proportional to f. This yields conditions described in 1.-3.

Function \(c_{p,\pm }\) is meromorphic. In the region \(|\text {Re}(\mu )| < \frac{1}{2}\) functions \(\frac{\omega }{\Gamma (1+\mu {\mp } \text {i}\lambda )}\) and \(\frac{\lambda {\mp } \text {i}\mu }{\Gamma (1 - \mu {\mp } \text {i}\lambda )}\) do not simultaneously vanish anywhere, while \(\frac{\Gamma (2 \mu +1)}{\Gamma (-2 \mu +1)}\) is holomorphic and nowhere vanishing. Hence, \(c_{p, \pm }\) is not of the indeterminate form \(\frac{0}{0}\) anywhere. \(\square \)

Let us note that eigenfunctions corresponding to real eigenvalues (which exist only for \(|\text {Im}(\lambda )| > \frac{1}{2}\)) decay at infinity only as fast as \(x^{- |\text {Im}(\lambda )|}\), not exponentially.

Consider a homogeneous operator \(D_p\) with \(p \in \mathcal {E}^\pm \) and its deformations \(D_{\omega ,\lambda }^f\), with f parametrized by \(\kappa \) so that \(D_{\omega ,\lambda }^f=D_p\) for \(\kappa =0\). Then for \(\kappa =0\) the point spectrum of \(D_{\omega ,\lambda }^f\) is \(\mathbb {C}_\pm \), but for every \(\kappa \ne 0\) it is disjoint from \(\mathbb {C}_\pm \).