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Sbornik: Mathematics, 2023, Volume 214, Issue 10, Pages 1415–1441
DOI: https://doi.org/10.4213/sm9861e
(Mi sm9861)
 

This article is cited in 1 scientific paper (total in 1 paper)

On eigenfunctions of the essential spectrum of the model problem for the Schrödinger operator with singular potential

M. A. Lyalinov

Saint Petersburg State University, St. Petersburg, Russia
References:
Abstract: We are concerned with generalized eigenfunctions of the continuous (essential) spectrum for the Schrödinger operator with singular $\delta$-potential that has support on the sides of an angle in the plane. Operators of this kind appear in quantum-mechanical models for quantum state destruction of two point-interacting quantum particles of which one is reflected by a potential barrier. We propose an approach capable of constructing integral representations for eigenfunctions in terms of the solution of a functional-difference equation with spectral parameter. Solutions of this equation are studied by reduction to an integral equation, with the subsequent study of the spectral properties of the corresponding integral operator. We also construct an asymptotic formula for the eigenfunction at large distances. For this formula a physical interpretation from the point of view of wave scattering is given.
Our approach can be used to deal with eigenfunctions in a broad class of related problems for the Schrödinger operator with singular potential.
Bibliography: 17 titles.
Keywords: essential spectrum, eigenfunctions, integral representation, functional-difference equation, asymptotics.
Funding agency Grant number
Russian Science Foundation 22-11-00070
This research was carried out with the financial support of the Russian Science Foundation (grant no. 22-11-00070), https://rscf.ru/en/project/22-11-00070/.
Received: 25.11.2022
Bibliographic databases:
Document Type: Article
MSC: 35J10, 35J25, 35P99
Language: English
Original paper language: Russian

§ 1. Introduction

1.1.

Jost (see [1]) considered a simplest model of one-dimensional scattering of two quantum particles interacting via a $\delta$-potential. Being scattered, one of the particles is reflected from an infinite wall so that the wave function $\Psi(\xi,\eta)$, which describes the state, satisfies the equation

$$ \begin{equation} \frac{\partial^2 \Psi}{\partial\xi^2}+ \frac{\partial^2 \Psi}{\partial\eta^2}+ \gamma\bigl[\theta(\xi)\delta(\eta)+\theta(\eta)\delta(\xi)\bigr] \Psi +E\Psi =0, \end{equation} \tag{1.1} $$
where $\theta(\,{\cdot}\,)$ is the Heaviside function, $\gamma>0$ is the coupling constant, and $E$ is the spectral parameter. The coordinates of particles are given by $\xi+\eta$ and $\xi-\eta$, and the potential in (1.1) is supported on the sides of a right angle (see Figure 1, a). We investigate the values of the energy $E\geqslant-\gamma^2/4$ and the solutions of equation (1.1) corresponding to the continuous (essential) spectrum. The generalized eigenfunction should also obey the antisymmetry condition $\Psi(\xi,\eta)=-\Psi(\eta,\xi) $. In the actual fact, here one speaks about the construction of the eigenfunction of the continuous spectrum of this operator with special singular potential and the description of the corresponding quantum scattering processes. This will be referred to as the Jost problem.

A single-layer potential type representation and some generalization of the Wiener-Hopf method (see [1]) were used to reduce the Jost problem to a functional-difference equation with meromorphic coefficient. In the case where the support of the singular potential lies on a right angle, a method for the solution of this equation was proposed in [2]; however, [2] deals neither with eigenfunctions nor with their asymptotic behaviour. The approach based on the Wiener-Hopf method can be used when the potential has support on the sides of a right angle; however, this method cannot be generalized to other related problems.

In the present paper we study eigenfunctions of the essential spectrum for a similar operator in the case where the potential has supported on points of the boundary of an angle of arbitrary opening $2\Phi$, $0<2\Phi<\pi$, in the plane1 (see Figure 1, b). Under this approach the antisymmetry condition on the eigenfunction is replaced by the equivalent condition that the eigenfunction is odd with respect to the axis $y=0$ (the Dirichlet condition on the boundary $y=0$). Next, we formulate the problem for the corresponding operator, in particular, in classical terms, including the equation, boundary conditions and the description of the class of functions.

Our main aim in this paper is to develop a method for the construction of eigenfunctions of the essential spectrum and study their asymptotic behaviour with respect to distance for this class of problems. The generalized Jost problem is used as a meaningful example in which we demonstrate the main ‘ingredients’ of our approach. In the actual fact, in what follows the approach worked out in [3]–[5] for a class of problems with singular potentials, to describe eigenfunctions of the discrete spectrum for a special class of Schrödinger operators with singular potential is adapted for generalized eigenfunctions of the essential (continuous) spectrum. Although in general we follow ideas from the above works, the technical implementation and corresponding estimates differ significantly. Our procedure for the construction of the eigenfunction is constructive and based on formulae. In addition, our approach depends substantially on the utilization of appropriate (Kontorovich-Lebedev or Watson-Bessel type) integral representations for the solution and a reduction to a functional-difference equation, and then, to an integral equation with spectral parameter; we also study the spectral properties of this integral equation. It turns out that to derive asymptotic formulae for the eigenfunction with respect to the distance one should use an alternative integral representation (a sort of the Sommerfeld integral representation). The corresponding asymptotic interpretation of generalized eigenfunctions allows one to interpret them ‘physically’ as outgoing and incoming surface waves localized near the support of the singular potential. Note that in problems of this type discrete spectrum can also exist to the left of the essential spectrum (see [4]).

It is worth pointing out that functional-difference equations provide a handy tool not only in applications to quantum theory (see [6], [7]) — they also work as key models in spectral theory (see, for example, [8] and the references there), in problems of acoustic and electromagnetic scattering (see [9]–[11]) and in the theory of wave oscillations of fluids (see [12]).

1.2. Notation and statement of the problem

Here we introduce the main objects of research and describe the self-adjoint operator and the corresponding spectral boundary-value problem in terms of the equation and boundary conditions. The self-adjoint Schrödinger operator $A_s$, which is considered in our paper, is defined by its sesquilinear form $a_s$ in $L_2(\mathbb R^2_+)$, which is semibounded, densely defined, and closed.2 A ray $l$ partitions the upper half-plane $\Omega =\mathbb R^2_+$, $(x,y)\in \Omega$, into two parts $\Omega_j$, $j=1,2$ (see Figure 1, b). In the polar coordinates $x=r\cos\varphi$, $y=r\sin\varphi$ we have $l=\{(r,\varphi)\colon r>0,\,\varphi=\Phi\}$, $0<\Phi<\pi/2$,

$$ \begin{equation*} \Omega_1=\bigl\{(r,\varphi)\colon r>0,\, 0<\varphi<\Phi\bigr\} \end{equation*} \notag $$
and
$$ \begin{equation*} \Omega_2=\bigl\{(r,\varphi)\colon r>0,\,\Phi<\varphi<\pi\bigr\}, \end{equation*} \notag $$
where $0<\Phi<\pi$. The corresponding quadratic form reads as
$$ \begin{equation*} a_s[U,U]=\int_{\Omega}\nabla U\cdot\overline{\nabla U} \,\mathrm{d}x-\gamma \int_{l}|U|^2 \,\mathrm{d} s, \end{equation*} \notag $$
where $\gamma>0$ is the Robin parameter, and $\operatorname{Dom}[a_s]= H^{1,0}(\Omega)$, $\Omega=\Omega_1\cup l\cup\Omega_2$ ($H^{1,0}(\Omega)$ is the Sobolev space of functions with Dirichlet condition on the boundary of the half-plane).

The operator $A_s$, which is generated by the form $a_s$, is self-adjoint and semibounded. It is realized3 as the Laplace operator with singular $\delta$-potential supported on $l$, that is, $A_s=-\triangle-\gamma \delta_l(x)$.

In what follows we only use the classical realization of the operator $A_s$ in terms of the equation and boundary conditions. For a quantum-mechanical interpretation of the operator $A_s$ for $\Phi=\pi/4$ and also for boundary conditions, see, for example, [2], § 1. We consider the equation

$$ \begin{equation*} A_s U = E U, \end{equation*} \notag $$
where $E$ is the spectral parameter. It is known (see, for example, [13]) that the essential spectrum $\sigma_e(A_s)$ of the operator $A_s$ is $[-\gamma^2/4,\infty)$, and in this paper we consider the case of $E\in [-\gamma^2/4,0)$. The generalized eigenfunctions for positive $E$ are constructed similarly but with some technical modifications.

We construct the classical solution $u=u_j$ in $\Omega_j$, $j=1,2$, satisfying the equations

$$ \begin{equation} \begin{gathered} \, -\triangle u_1(r,\varphi)- Eu_1(r,\varphi)=0 ,\qquad (r,\varphi)\in \Omega_1, \\ -\triangle u_2(r,\varphi)- Eu_2(r,\varphi)=0 ,\qquad (r,\varphi)\in \Omega_2, \end{gathered} \end{equation} \tag{1.2} $$
and the boundary conditions4
$$ \begin{equation} u|_{y=0} =0, \end{equation} \tag{1.3} $$
$$ \begin{equation} \frac{\partial u_1}{\partial n}\bigg|_{l} -\frac{\partial u_2}{\partial n}\bigg|_{l}= \gamma u_1|_{l}\quad\text{and} \quad u_1|_{l}=u_2|_{l}, \end{equation} \tag{1.4} $$
where the unit normal vector $n$ on $l$ is directed inside $\Omega_2$. In view of the condition $u\in H^{1,0}(\Omega)$, it can be expected that
$$ \begin{equation} u_j(r,\varphi)= O(r^{\delta_*}), \qquad \delta_*>0,\quad r\to 0, \end{equation} \tag{1.5} $$
uniformly with respect to $\varphi$. We will show that a nontrivial solution of the problem (an eigenfunction of the continuous spectrum) (1.2)(1.5) exists for almost all $E\in [-\gamma^2/4,0)$; in addition, it is not square-integrable because of its behaviour as ${r\to \infty}$. To this end we find the asymptotic behaviour of the solution as $r\to \infty$.

1.3. The structure of the paper and the results

In § 2, the Watson-Bessel integral representations are used for incomplete separation of variables.5 The unknown functions in the integrand are chosen so as to satisfy both the equations and the boundary condition. In particular, the continuity and Robin boundary conditions (1.4) lead to a functional-difference equation with spectral (characteristic) parameter related to $E$ directly. The resulting homogeneous functional-difference equation and its meromorphic solutions from a special class play a fundamental role in the construction of eigenfunctions of the continuous spectrum.

In § 3 we reduce the functional-difference equation to an integral equation with self-adjoint operator $\mathbf{K}=\mathbf{M}+\mathbf{V}$, which is a compact perturbation $\mathbf{V}$ of the so-called Mehler operator $\mathbf{M}$, whose spectral properties are known (see, for example, [5]): the essential (in the actual fact, continuous and simple) spectrum $\sigma_e(\mathbf{M})$ is the interval $[0,1]$. From Weyl’s compact perturbation theorem it follows that the spectrum $\sigma_e(\mathbf{K})$ is $[0,1]$. Solutions of the functional-difference equation in the corresponding class are constructed in terms of generalized eigenfunctions of the operator $\mathbf{K}$. This gives us an integral representation for a generalized eigenfunction of the operator $A_s$.

Toward this end in view, in § 4 we consider the behaviour of the generalized eigenfunction of the operator $\mathbf{K}$ in a neighbourhood of a singular point of the kernel. We find the leading singularity, at which the generalized eigenfunction is not square-integrable (which is natural). We derive an integral equation for the (square-summable) correction. The corresponding integral equation, showing an analytic dependence on the spectral parameter, and its solvability are examined using the machinery of the analytic Fredholm alternative. The main technical challenges here are related to asymptotic estimates for the behaviour of the resolvent kernel of the Mehler operator on the continuous spectrum (see § 7).

In § 5 the results obtained are used to construct the solution of the functional-difference equation in the required class, for almost all values of the characteristic (spectral) parameter in the equation. We also discuss the behaviour of the solution on the complex plane and its asymptotical behaviour away from the real axis.

The Watson-Bessel integral representations are ill suited for constructing asymptotic formulae for the generalized eigenfunction of the operator $A_s$ at large distances. To circumvent this difficulty, in § 6 we invoke Sommerfeld-type integral representations. The integrand, which is related directly to the above solution of the functional-difference equation by means of the Fourier transform along the imaginary axis, is shown to solve the system of (Malyuzhinets’s) functional equations in the special class of meromorphic functions. The use of these equations proves to be instrumental in finding the singularities (poles) of the integrand in the Sommerfeld integral and in obtaining an asymptotic formula for the integral using the saddle-point method. The leading term of the asymptotic formula depends on the poles closest to the imaginary axis, and the correction term is a function of the other poles or of the saddle points. The correction terms are known to be exponentially small with respect to the distance, whereas the leading term (an exponential) is only bounded. The leading term can be interpreted physically as a surface wave travelling towards infinity (or from infinity) along a ray $l$. For almost all $E\in [-\gamma^2/4,0)$ the generalized eigenfunction of the operator $A_s$ is bounded on the support of the singular potential and is exponentially small for each direction outside $l$ as $r\to\infty$.

§ 2. Watson-Bessel integral representations for the solution and incomplete separation of variables

Here and in what follows we precede the main results by motivations for them and by underlying calculations.

In contrast to the situation with integral representations for eigenfunctions of the discrete spectrum (see [4]), where the Kontorovich-Lebedev integral representation are used, for the essential spectrum it is natural to adhere to representations of the form6

$$ \begin{equation} \begin{gathered} \, u_1(r,\varphi)=\mathrm{i}\int_{C_0^b} \exp\biggl(-\frac{\mathrm{i}\pi\nu}2\biggr)J_\nu(\mathrm{i}\kappa r)\frac{\sin(\nu\varphi)}{\sin(\nu\Phi)} H_1(\nu)\,\mathrm{d}\nu, \qquad \varphi\in [0,\Phi], \\ u_2(r,\varphi)= \mathrm{i}\int_{C_0^b} \exp\biggl(-\frac{\mathrm{i}\pi\nu}2\biggr)J_\nu(\mathrm{i}\kappa r)\frac{\sin(\nu\overline{\varphi} )}{\sin(\nu\overline{\Phi})} H_2(\nu)\,\mathrm{d}\nu, \qquad \varphi\in [\Phi,\pi], \end{gathered} \end{equation} \tag{2.1} $$
for which the rapid convergence of the integrals can be secured. Here $\overline{\Phi}=\pi-\Phi$, $\overline{\varphi} =\pi-\varphi$, $\kappa=\sqrt{-E}$, $J_\nu(\,{\cdot}\,)$ is the Bessel function, and the functions $H_{1,2}(\,{\cdot}\,)$ must be evaluated based on boundary conditions (1.4). The contour $C_0^b= (\infty- \mathrm{i}b, -\mathrm{i}b)\cup[-\mathrm{i}b, \mathrm{i}b]\cup (\mathrm{i}b, \mathrm{i}b+\infty)$, $b>0$, is shown in Figure 2. Representations (2.1) can be shown to converge rapidly and uniformly with respect to $(r,\varphi)$ due to the rapid decay of the Bessel function on the contour (this is discussed below). They also satisfy equations (1.2) in the classical sense because of the equality
$$ \begin{equation*} \begin{aligned} \, &(\kappa r)^2\biggl\{\frac{d^2}{d(\kappa r)^2}+\frac{1}{\kappa r}\frac{d}{d\kappa r}-\biggl(1+\frac{\nu^2}{(\kappa r)^2}\biggr) \biggr\}J_\nu(\mathrm{i}\kappa r)u_\nu(\varphi) \\ &\qquad\qquad +\biggl(\frac{d^2}{d\varphi^2}+\nu^2\biggr)u_\nu(\varphi)J_\nu(\mathrm{i}\kappa r)=0, \end{aligned} \end{equation*} \notag $$
where $u_\nu(\varphi)=\cos(\nu\varphi)$ or $u_\nu(\varphi)=\sin(\nu\varphi)$. The Dirichlet condition (1.3) is satisfied by taking $u_\nu(\varphi)=\sin(\nu\varphi)$ ($\sin(\nu\overline{\varphi})$, respectively) under the integral sign.

Prior to substituting Watson-Bessel integrals into the boundary conditions (1.4) and deriving functional-difference equations for the calculation of $H_{1,2}$, it is handy to describe the class of functions for which these equations have solutions and the integral representations (2.1) solve the problem in the classical sense. Thus, they can be used to produce the generalized eigenfunction of the essential spectrum. On the basis of our experience on our experience in constructing eigenfunctions of the discrete spectrum [4], [5], we introduce the class $\mathcal M$ of meromorphic functions $h$ such that

$\bullet$ $h(\nu)= -h(-\nu)$ is odd (or $h(\nu)= h(-\nu)$ is even);

$\bullet$ $h$ is holomorphic in $\Pi_{\delta}:=\Pi(-\delta,\delta)=\{\nu\in \mathbb C\colon -\delta<\operatorname{Re} \nu< \delta\}$ for some $\delta>0$, that is, in a neighbourhood of the imaginary axis $\mathrm{i}\mathbb R$. All the poles of $h$ lie in the strip $|\operatorname{Im} \nu| <b$ for some positive $b$;

$\bullet$ $ |h(\nu)|<\mathrm{Const}|\exp(-|\nu|(\pi/2) \sin|\psi|)|$, $|\nu|\to\infty$ (this estimate holds for all $\pm\psi\in [0,\pi/2]$, for $\nu= |\nu|\exp(\mathrm{i}\psi)$, $|\nu|\to\infty$ outside the strip $|\operatorname{Im} \nu| <b$);

$\bullet$ $d(\nu+1)$ is holomorphic in the strip $\Pi(-1,0)$ and continuous up to the boundary, and $ d(\nu-1)$ is holomorphic in the strip $\Pi(0,1)$ and continuous up to the boundary, where $d(\nu)=h(\nu)(\cot\nu\Phi +\cot\nu\overline{\Phi})$.

Let us discuss the convergence of the Watson-Bessel integrals (2.1) for functions $H_{1,2}$ in the class $\mathcal M$. To this end we use an asymptotic formula for Bessel functions ($|\nu|\gg 1$, $|{\arg\nu}|\leqslant\pi/2$, $|\arg z|<\pi$):

$$ \begin{equation*} J_\nu(z)\sim \frac{[z/2]^\nu}{\Gamma(\nu+1)} = \biggl[\frac z2\biggr]^\nu \frac{\exp\{-\nu[\log\nu-1]\}}{\sqrt{2\pi \nu} }\biggl(1+O\biggl(\frac1\nu\biggr)\biggr) \end{equation*} \notag $$
on the contour $C_\psi^b=(\mathrm{e}^{-\mathrm{i}\psi}\infty-\mathrm{i}b, -\mathrm{i}b)\cup[-\mathrm{i}b, \mathrm{i}b]\cup (\mathrm{i}b, \mathrm{i}b+\mathrm{e}^{\mathrm{i}\psi}\infty)$, where $\psi\in [0,\pi/2]$ and $C_0^b=C_{\psi=0}^b$. In what follows we must be able to deform the contour $C_0^b$ into $C_{\pi/2}^b= \mathrm{i}\mathbb R$ (that is, the imaginary axis). On the contour $C_\psi^b$, for $\nu=|\nu|\mathrm{e}^{\mathrm{i}\psi}\to\infty$ (see Figure 3) the integrand in (2.1) is estimated as follows:
$$ \begin{equation} \begin{aligned} \, \notag &\biggl|\exp\biggl(-\frac{\mathrm{i}\pi\nu}2\biggr)J_\nu(\mathrm{i}\kappa r)\frac{\sin(\nu\varphi)}{\sin(\nu\Phi)} H_1(\nu)\biggr| \\ &\quad\leqslant \notag \frac{C}{\sqrt{|\nu|}} \biggl|\frac{ \exp(-\mathrm{i}\pi\nu/2)\exp(-\nu[\log\nu-1])+\nu\log(\mathrm{i}\kappa r/2))}{\exp(|\nu|({\pi}/{2}) \sin|\psi|)}\biggr| \\ &\quad\leqslant \frac{C}{\sqrt{|\nu|}} \biggl|\exp\biggl\{-|\nu|\biggl[\log|\nu|-1 -\log\frac{\kappa r}2\biggr]\cos\psi+ |\nu|\biggl[\psi\sin\psi -\frac\pi2\sin|\psi|\biggr] \biggr\}\biggr|, \end{aligned} \end{equation} \tag{2.2} $$
where we have used Stirling’s formula for the gamma function. A similar estimate also holds for the second integrand in (2.1). In view of estimate (2.2) for $\psi=0$, we have the following result.

Lemma 1. Let $H_{1,2}\in \mathcal M$. Then the Watson-Bessel integral representations (2.1) for $u_{1,2}$ converge absolutely and uniformly with respect to $(r,\varphi)$ on any compact subset of $\Omega_{1,2}$, are twice continuously differentiable7 with respect to $(r,\varphi)$ in $\Omega_{1,2}$, satisfy equations (1.2) and condition (1.3), and have continuous derivatives at all smooth points of the boundary (that is, everywhere except, possibly, the corner point $O$).

Using Lemma 1 one can substitute the Watson-Bessel integral representations into the equation and the boundary conditions and verify them in the classical sense.

Remark 1. By applying estimate (2.2) and the properties of $\mathcal M$-functions one can deform the contour $C_0^b$ into $C_{\psi}^b$, and, in particular, into the imaginary axis ($\psi= \pi/2$) with preservation of the convergence of the integrals (see Figure 3).

2.1. Boundary condition on $l$ and a functional-difference equation for $H_{1,2}$

The continuity condition in (1.4) implies that

$$ \begin{equation} H_1(\nu)=H_2(\nu). \end{equation} \tag{2.3} $$
From the Robin-type condition in (1.4) we have
$$ \begin{equation*} \begin{aligned} \, &\frac{1}{\kappa r}\biggl(\frac{\partial u_1}{\partial \varphi}- \frac{\partial u_2}{\partial \varphi}\biggr)\bigg|_{\varphi=\Phi}- \frac{\gamma}{2\kappa}( u_1+u_2)|_{\varphi=\Phi} \\ &\qquad =\mathrm{i}\int_{C_0^b}\mathrm{d}\nu \exp\biggl(-\frac{\mathrm{i}\pi\nu}2\biggr)\mathrm{i}\nu\frac{J_\nu(\mathrm{i}\kappa r)}{\mathrm{i}\kappa r}\bigl\{ H_1(\nu) \cot(\nu\Phi) + H_2(\nu) \cot(\nu\overline{\varphi} ) \bigr\} \\ &\qquad\qquad -\frac{\gamma}{2\kappa}\mathrm{i}\int_{C_0^b}\mathrm{d}\nu \exp\biggl(-\frac{\mathrm{i}\pi\nu}2\biggr) (H_1(\nu)+H_2(\nu))J_\nu(\mathrm{i} \kappa r)=0. \end{aligned} \end{equation*} \notag $$

In view of equality (2.3) it is convenient to introduce the new unknown

$$ \begin{equation} D(\nu)= H_1(\nu) \cot(\nu\Phi) + H_2(\nu) \cot(\nu\overline{\Phi}) =H_1(\nu) (\cot(\nu\Phi) + \cot(\nu\overline{\Phi}) ). \end{equation} \tag{2.4} $$
Employing the relation $2\nu J_\nu(z)= z(J_{\nu+1}(z)+J_{\nu-1})$ for the Bessel functions (see formula 8.471 in [14]) we obtain
$$ \begin{equation*} \begin{aligned} \, &\frac{\mathrm{i}}{2}\int_{C_0^b}\mathrm{d}\nu \exp\biggl(-\frac{\mathrm{i}\pi\nu}2\biggr)\mathrm{i}D(\nu)\bigl(J_{\nu+1}(\mathrm{i} \kappa r) +J_{\nu-1}(\mathrm{i} \kappa r)\bigr) \\ &\qquad-\frac{2\gamma}{\kappa}\, \frac{\mathrm{i}}{2}\int_{C_0^b}\mathrm{d}\nu \exp\biggl(-\frac{\mathrm{i}\pi\nu}2\biggr) \frac{D(\nu)}{\cot(\nu\Phi) + \cot(\nu\overline{\varphi})} J_\nu(\mathrm{i} \kappa r)=0. \end{aligned} \end{equation*} \notag $$
We split the first integral into two terms with $J_{\nu+1}$ and $J_{\nu-1}$ and change the integration variable, respectively, as follows: $\nu+1\to \nu$ and $\nu-1\to \nu$. The integration contour $C_0^b$ is transformed into $C_0^b+1$ and $C_0^b-1$, respectively, as shown in Figure 2. As a result, we obtain
$$ \begin{equation*} \begin{aligned} \, &\frac{\mathrm{i}}{2}\int_{C_0^b+1}\mathrm{d}\nu \exp\biggl(-\frac{\mathrm{i}\pi[\nu-1]}2\biggr)\mathrm{i}D(\nu-1)J_{\nu}(\mathrm{i} \kappa r) \\ &\qquad-\frac{\mathrm{i}}{2}\int_{C_0^b-1}\mathrm{d}\nu \exp\biggl(-\frac{\mathrm{i}\pi[\nu+1]}2\biggr)\mathrm{i}D(\nu+1)J_{\nu}(\mathrm{i} \kappa r) \\ &\qquad-\frac{2\gamma}{\kappa}\frac{\mathrm{i}}{2}\int_{C_0^b}\mathrm{d}\nu \exp\biggl(-\frac{\mathrm{i}\pi\nu}2\biggr) \frac{D(\nu)}{\cot(\nu\Phi) + \cot(\nu\overline{\varphi} )}J_\nu(\mathrm{i} \kappa r)=0. \end{aligned} \end{equation*} \notag $$
Recalling that $H_{1,2}$ lies in the class $\mathcal M$, we deform the contours ${C_0^b\pm 1}$ into ${C_0^b}$. This gives us
$$ \begin{equation*} \frac{\mathrm{i}}{2}\int_{C_0^b}\mathrm{d}\nu \exp\biggl(-\frac{\mathrm{i}\pi\nu}2\biggr)\biggl(D(\nu+1) -D(\nu-1)-\frac{2\gamma}{\kappa}\,\frac{D(\nu)}{\cot(\nu\Phi) +\cot(\nu\overline{\Phi})}\biggr)J_\nu(\mathrm{i} \kappa r)=0. \end{equation*} \notag $$
The last equality, and therefore the Robin-type boundary condition, is satisfied if $D\in \mathcal M$ is a solution of the functional-difference equation
$$ \begin{equation} D(\nu+1)-D(\nu-1)-2\mathrm{i} \Lambda W(\nu) D(\nu)=0, \end{equation} \tag{2.5} $$
where $\Lambda={\gamma}/({2\kappa})$ and
$$ \begin{equation*} W(\nu)=\frac{-2\mathrm{i}}{\cot(\nu\Phi) +\cot(\nu\overline{\Phi})} \end{equation*} \notag $$
is called a meromorphic potential. Using the above observations and Lemma 1 we arrive at the following result.

Proposition 1. Let there exist a nontrivial solution $D$ of equation (2.5) for some $\Lambda\geqslant 1$, and let $H_{1,2}$ in (2.4) belong to the class $\mathcal M$. Then the Watson-Bessel integral representations (2.1) satisfy equations (1.2) and the boundary conditions (1.3), (1.4) in the classical sense.

Note that estimate (1.5) can be verified by deforming the contour $C_0^b$ to the right so that the real axis intersects it at the point $\nu=\delta_*>0$, and by replacing the Bessel function under the integral sign by an asymptotic formula for it as $r\to 0$. This is possible because of the rapid convergence of the integral.

It is clear that the investigation of nontrivial solutions of equation (2.5) in the required class of meromorphic functions is the main problem. The construction of the generalized eigenfunction will be complete once we have found the asymptotic behaviour of $u_{1,2}$ as $r\to \infty$. This is done by a transition to the Sommerfeld integral representation.

§ 3. Reducing the functional-difference equation (2.5) to an integral equation and examining its properties

Writing equation (2.5) in the form $D(\nu+1)- D(\nu- 1)=2\mathrm{i} \Lambda W(\nu) D(\nu)$ and applying to it Lemma 3.1 in [5] we obtain the integral representation8

$$ \begin{equation} D(\nu)=- \frac{\Lambda}{2}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty} d\tau \frac{W(\tau)\sin\pi\tau}{\cos\pi\tau+\cos\pi\nu}D(\tau), \qquad \nu\in \Pi_{1+\delta}. \end{equation} \tag{3.1} $$
Note that the solution $D(\,{\cdot}\,)$ is even. Let us discuss representation (3.1). If $D(\,{\cdot}\,)$ on the right-hand side is known and integrable on the imaginary axis, then the right-hand side of the representation is holomorphic in the strip $\Pi_{1}$. This follows from the fact that the denominator in the integrand vanishes in the strip at the points $\tau=\nu\pm 1$ closest to the imaginary axis. The function $W(\,{\cdot}\,)$ is holomorphic in the neighbourhood of width $2\delta$ of the imaginary axis, and so, by deforming the contour in this neighbourhood, we conclude that the left-hand side is holomorphic in the strip $ \Pi_{1+\delta}$. The meromorphic extension of $D(\,{\cdot}\,)$ onto the entire complex plane is obtained by using the functional-difference equation. It is easily seen that the singularities of the meromorphic extension lie on the real axis because the singularities of the potential are real and since the shifts in the functional-difference equation are made along the real axis. So, if for $\Lambda \geqslant 1$ we can find the values of $D(\,{\cdot}\,)$ on the imaginary axis with a ‘regular’ behaviour at infinity, then $D(\,{\cdot}\,)$ extends meromorphically to $\mathbb C$. Setting $\nu\to \mathrm{i}\mathbb R$, we arrive at the integral equation (3.1).

Proposition 2. Assume that for $\Lambda\geqslant 1$ the integral equation (3.1) has a solution $D(\,{\cdot}\,)$ which is integrable on the positive part of the imaginary axis $\mathrm{i}\mathbb R_+$ and satisfies (for $\nu\to\pm\mathrm{i}\infty$) the estimate

$$ \begin{equation*} |D(\nu)|<\mathrm{Const}\biggl|\exp\biggl(-\frac{\pi|\nu|}2 \biggr)\biggr| \end{equation*} \notag $$
on the imaginary axis. Then for $\Lambda\geqslant 1$ equation (2.5) has a nontrivial even meromorphic solution $D$. In addition, this solution has singularities on the real axis and satisfies the above estimate in the strip $\Pi_{1+\delta}$.

Let us now verify that the integral equation (3.1) has a solution with the properties required in Proposition 2. It will be useful and convenient to transform this equation (3.1) into an integral one with self-adjoint operator. Since the integrand is even, we will obtain an equation on the half-axis. Let us introduce the new variables

$$ \begin{equation*} x= \frac{1}{\cos\pi\nu}, \qquad y= \frac{1}{\cos\pi t}\quad\text{and} \quad \frac{\mathrm{d}y}{\pi}= \frac{\sin\pi t}{\cos^2\pi t}\,\mathrm{d}t \end{equation*} \notag $$
and the unknown
$$ \begin{equation*} h(x)= \cos\pi\nu D(\nu)|_{x= 1/\cos(\pi\nu)}, \end{equation*} \notag $$
$x,y\in [0,1]$. We have
$$ \begin{equation} h(x)-\frac{\Lambda}{\pi}\int_{0}^{1}\mathrm{d}y\frac{w_0(y)}{x+y} h(y)=0, \end{equation} \tag{3.2} $$
where
$$ \begin{equation*} w_0(y)=W(t)|_{y=1/\cos(\pi t)}>0 \end{equation*} \notag $$
and $w_0(y)= 1 +o(1)$ as $y\to 0$. Note that $W(t)$ is odd, positive on the positive part of the imaginary axis, $W(t)= 1 +O(\exp(-d_* |\nu|))$, $d_* = 2 \min (\Phi,\overline{\Phi}) = \Phi$ as $\operatorname{Im} \nu\to\infty$, $\operatorname{Re}\nu=0$, and $W$ is meromorphic. From (3.2) we obtain the integral equation with symmetric kernel
$$ \begin{equation} \rho(x)-\frac{\Lambda}{\pi}\int_{0}^{1}\mathrm{d}y\frac{ w(x,y)}{x+y} \rho(y)=0, \end{equation} \tag{3.3} $$
where $\rho(x)=\sqrt{w_0(x)}h(x)$ and $w(x,y)=\sqrt{w_0(x)w_0(y})$.

Introducing the characteristic parameter $\Lambda$ and the spectral parameter $\mu=\Lambda^{-1}$, we write equation (3.3) in the form

$$ \begin{equation} (\mathbf{K}\rho)(x)=\mu\rho(x) \end{equation} \tag{3.4} $$
in $L_2([0,1])$.

Let us study the properties of the operator $\mathbf{K}$ with symmetric kernel, which can be represented as the sum (see [5])

$$ \begin{equation} \frac{w(x,y)}{x+y}=\frac{ 1}{x+y} +\frac{ v(x,y)}{x+y}, \end{equation} \tag{3.5} $$
$v(x,y)= w(x,y)-1=\mathrm{O}(x^{b_*}+y^{b_*})$ as $(x,y)\to (0,0)$, where $ b_*=d_*/\pi < 1$. The kernel $v(x,y)/(x+y)$ is square integrable.

Thus we have the following result.

Lemma 2. The operator $\mathbf{K}\colon L_2([0,1])\to L_2([0,1])$ in (3.4) is bounded and self-adjoint. In addition, $\mathbf{K}$ is positive and can be written as

$$ \begin{equation} \mathbf{K}=\mathbf{M}+\mathbf{V} \end{equation} \tag{3.6} $$
in accordance with the representation (3.5) of the kernel, where $\mathbf{M}$ is the so-called Mehler operator defined in $L_2([0,1])$ by
$$ \begin{equation*} (\mathbf{M}{r})(x)=\frac{1}{\pi}\int_{0}^{1}\frac{\mathrm{d} y}{x+y}\, \mathbf{r}(y). \end{equation*} \notag $$
The integral operator $\mathbf{V}$ in (3.6) is defined in $ L_2([0,1])$ by
$$ \begin{equation*} (\mathbf{V}\rho)(x)=\frac{1}{\pi}\int_{0}^{1} \mathrm{d} y\, \frac{v(x,y)}{x+y}\rho(y) \end{equation*} \notag $$
and is an operator in the Hilbert-Schmidt class $S_2 $ (this follows from the properties of the function ${v}(x,y)$).

The essential spectrum of the operator $\sigma_e(\mathbf{K})$ is the closed interval $\mu\in [0,1]$ (or $\Lambda=\mu^{-1}\geqslant 1$).

Indeed, it is known that the spectrum of the Mehler operator $\mathbf{M}$ is essential (absolutely continuous and simple) and coincides with the interval $ [0,1]$. That $\mathbf{M}$ is diagonalizable follows from the well-known Mehler formulae (see [15] and also § 4.1 in [5]). The operator $\mathbf{V}$ is compact, hence by Weyl’s theorem the essential spectrum is preserved under this perturbation. Unlike [5], here we do not discuss the possible existence of discrete spectrum to the right of $\mu=1$ ($\mathbf{K}\geqslant 0$). Instead, we are concerned with eigenfunctions of the essential (continuous) spectrum $\sigma_e(\mathbf{K})$ for $\mu\in (0,1]$.

§ 4. Integral equation and estimate for the solution for $\mu\in (0,1]$

We parameterize the points $\mu$ on the interval $[0,1]$ by a parameter $p\in [0,\infty)$ in accordance with the equality $\mu(p)={1}/{\cosh(\pi p)}$, $p(\mu)=({1}/{\pi})\log(1/\mu+\sqrt{1/\mu^2\!-\!1})$. A branch of $p(\,{\cdot}\,)$ in the plane cut along $[0,1]$ is fixed as in [5], § 4.2. Consider the function $\sqrt{\mu^2-1}$ in the complex plane cut along $[-1,1]$, where the branch is fixed by the condition $\sqrt{\mu^2-1}>0$ for $\mu>1$. Note that the function $(1-\mathrm{i}\sqrt{\mu^2-1})/{\mu}$ does not take positive values. So we can assume that $\arg((1-\mathrm{i}\sqrt{\mu^2-1})/{\mu})\in (0,2\pi)$. As a result, the function

$$ \begin{equation*} p(\mu):=\frac{1}{\pi}\log\biggl(\frac{1-\mathrm{i}\sqrt{\mu^2-1}}{\mu}\biggr) \end{equation*} \notag $$
is holomorphic in $\mathbb{C}\setminus [-1,1]$.

The eigenfunctions of the continuous spectrum of the operator $\mathbf{M}$ are known to be

$$ \begin{equation*} \mathcal P_p(x):= \frac{\sqrt{p\tanh(\pi p)}}{x} P_{\mathrm{i}p-1/2}\biggl(\frac1x\biggr) \end{equation*} \notag $$
(see, for example, § 4.1 in [5]; the corresponding results follow from Mehler’s formulae [15]); here the $P_{\mathrm{i}p-1/2}(\,{\cdot}\,)$ are Legendre functions (see [14], 8.715(1), and § 7), which have the integral representation
$$ \begin{equation*} P_{\mathrm{i}\tau-1/2}(\cosh\alpha)= \frac{\sqrt{2} }{\pi}\int_0^\alpha\frac{\cos(\tau t) \,\mathrm{d}t}{\sqrt{\cosh\alpha-\cosh t}}. \end{equation*} \notag $$
An asymptotic formula for eigenfunctions follows from formula 8.772(1) in [14]:
$$ \begin{equation*} \begin{aligned} \, \mathcal P_p(x) &= \frac{\sqrt{p\tanh(\pi p)}}{x} \biggl(\frac{\Gamma(-\mathrm{i}p)} {\Gamma(-\mathrm{i}p+1/2)}\biggl[\frac{x}{2}\biggr]^{1/2+\mathrm{i}p} \\ &\qquad +\frac{\Gamma(\mathrm{i}p)}{\Gamma(\mathrm{i}p+1/2)} \biggl[\frac{x}{2}\biggr]^{1/2-\mathrm{i}p}\biggr)\biggl(\frac{1}{\sqrt{\pi}}+O(x^2)\biggr) \end{aligned} \end{equation*} \notag $$
as $x\to 0+$ for $p> 0$, and $\mathcal P_p(x)=O(1)$ as $p\to\infty$ for $1\geqslant x>0$. The functions $\mathcal P_p(x)$ are real for $p\geqslant 0$. The generalized eigenfunction $\mathcal P_p(\,{\cdot}\,)$ is, as expected, a solution of the equation
$$ \begin{equation} (\mathbf{M}\mathcal P_p)(x)=\mu(p)\mathcal P_p(x) \end{equation} \tag{4.1} $$
and does not lie in $ L_2([0,1])$ (in view of its behaviour near the point $x=0$).

It seems natural to conjecture that the generalized eigenfunction of the operator $\mathbf{K}$ displays a similar asymptotic behaviour near the point $x=0$, as the operators $\mathbf{K}$ and $\mathbf{M}$ differ by a compact operator. Our aim in this section is to show that the generalized eigenfunction $\rho(\,{\cdot}\,)$ of the operator $\mathbf{K}$ differs from $\mathcal P_p(\,{\cdot}\,)$ by an $ L_2([0,1])$-function, that is,

$$ \begin{equation} \rho(x)-\mathcal P_p(x)=:g_p(x), \quad p\geqslant 0, \qquad g_p\in L_2([0,1]). \end{equation} \tag{4.2} $$
To prove this, we find an equation for $g_p\in L_2([0,1])$. Subtracting (4.1) from (3.4) we obtain
$$ \begin{equation} [\mathbf{M}-\mu(p)\mathbf{I}]g_p+\mathbf{V}g_p =-\mathbf{V}\mathcal P_p, \qquad p\geqslant 0. \end{equation} \tag{4.3} $$

Recall some properties of the resolvent $[\mathbf{M}-\mu(p)\mathbf{I}]^{-1}$ of the operator $\mathbf{M}$ (see, for example, § 4.2 in [5]). This resolvent is holomorphic in $\mathbb C\setminus [0,1]$ and has the representation

$$ \begin{equation} u(x)=[\mathbf{M}-\mu \mathbf{I}]^{-1}f(x)=-\frac{1}{\mu}\{\mathbf{I}+ \mathbf{A}_\mu\}f(x), \end{equation} \tag{4.4} $$
where
$$ \begin{equation} [\mathbf{A}_\mu f](x)= \frac{1}{\pi} \int_0^1 a(x,y;\mu) f(y) \,\mathrm{d}y \end{equation} \tag{4.5} $$
is an integral operator with kernel
$$ \begin{equation} a(x,y;\mu)= \pi\int_0^\infty\frac{\mathcal P_p(x)\mathcal P_p(y)}{ \mu\cosh(\pi p)-1}\,\mathrm{d}p . \end{equation} \tag{4.6} $$
The resolvent kernel has limit values on the sides of the cut along the spectrum $[0,1]$:
$$ \begin{equation*} a_\pm(x,y;s):=\lim_{\epsilon\to 0+} a(x,y;s\pm {\mathrm{i} \epsilon}) , \qquad s\in (0,1), \end{equation*} \notag $$
and at the endpoint $\mu=1$ (see [5], § 4.2). Let $\mathcal B_1$ be the disc of radius 1 with centre $\mu=1$ cut along the spectrum. At each $\mu$ in $\mathcal B_1$, including the sides of the cut and the point $\mu=1$, we have the estimate
$$ \begin{equation*} |a(x,y;\mu)|\leqslant C\frac{|\log (2/x) \log(2/ y)|}{\sqrt{xy}}, \qquad (x,y)\in (0,1]\times(0,1]. \end{equation*} \notag $$
This estimate follows from (4.6); however, for further purposes we will also require the asymptotics of the kernel as $(x,y)\to (0,0)$ close to the cut $[0,1]$ and at its endpoints.

The resolvent operator (4.4), (4.5) defines an isomorphism of the space $ L_2([0,1])$ for all $\mu$ outside the spectrum. Consider the open rectangle $\omega_+$ of small height $\epsilon_0$ in the upper half-plane ($\mu\in \mathbb C_+$) whose base $Q_+$ coincides with the upper side of the cut along $(0,1)$. Applying the resolvent to the left-hand side of (4.3), for $\mu \in \omega_+ $ we have

$$ \begin{equation} g_p+ [\mathbf{M}-\mu\mathbf{I}]^{-1}\mathbf{V}g_p =- [\mathbf{M}-\mu\mathbf{I}]^{-1}\mathbf{V}\mathcal P_p. \end{equation} \tag{4.7} $$
From the estimate $v(x,y)= O(x^{b_*}+y^{b_*})$ as $(x,y)\to (0,0)$ it easily follows that $\mathbf{V}\mathcal P_p\in L_2(0,1)$. Noting that $\mu \in \omega_+ $ and the operator $\mathbf{B}(\mu):=[\mathbf{M}-\mu\mathbf{I}]^{-1}\mathbf{V}$ is analytic with respect to these $\mu$, let us apply the analytic Fredholm alternative (see, for example, [16], Ch. 1, § 8, Theorem 2). The operator $\mathbf{B}(\mu)$ is compact for all $\mu \in \omega_+ $, and for some $\mu_1 \in \omega_+ $ (for example, if the absolute value of $\mu_1 $ is sufficiently large) the bounded inverse $[\mathbf{I}+\mathbf{B}(\mu) ]^{-1}$ exists. Hence there exists a meromorphic operator-valued function $[\mathbf{I}+\mathbf{B}(\mu) ]^{-1}$ in $\omega_+ $. Consequently, the operator $\mathbf{I}+\mathbf{B}(\mu) $ is boundedly invertible for all $\mu \in \omega_+ $, with the possible exception of a discrete set $\mathcal N$ of poles. Hence we have the following result.

Lemma 3. Equation (4.7) has a unique solution $g_p$ in $ L_2([0,1])$ for $\mu(p)\in \omega_+ \setminus \mathcal N $.

However, we are interested in the solvability of (4.7) on the lower part $Q_+$ of the boundary of the rectangle $\omega_+$, that is, on the boundary of the domain of analyticity, the spectrum. We invoke Theorem 3 in [16], Ch. 1, § 8, which describes the properties of an analytic operator-valued function on the boundary of its domain of analyticity. By the above, in order to carry over the conclusion of Lemma 3 to the values $\mu\in\omega_+ \cup Q_+\setminus \mathcal N_+$, where $\mathcal N_+$ is some Lebesgue nullset, we need to verify that the operator-valued function $\mathbf{B}(\,{\cdot}\,)$ is norm-continuous up to $Q_+$, that is,

$$ \begin{equation} \|\mathbf{B}(\mu+\mathrm{i}\epsilon)-\mathbf{B}(\mu)\|\to 0, \qquad \epsilon\to 0, \end{equation} \tag{4.8} $$
and $\mu\in Q_+=(0,1)$. To do this we estimate $\|\mathbf{B}(\mu+\mathrm{i}\epsilon)-\mathbf{B}(\mu)\|$ as follows:
$$ \begin{equation*} \begin{aligned} \, &\|(\mathbf{B}(\mu+\mathrm{i}\epsilon)-\mathbf{B}(\mu))\rho\|^2= \int_0^1 \biggl|(\mathbf{B}(\mu+\mathrm{i}\epsilon)-\mathbf{B}(\mu))\rho(x)\biggr|^2 \,\mathrm{d} x \\ &\leqslant\mu^{-1}\int_0^1 \mathrm{d} x\biggl| \frac{1}{\pi}\int_0^1 \mathrm{d} z |a(x,z;\mu+\mathrm{i}\epsilon)-a(x,z;\mu)| \biggl(\frac{1}{\pi}\int_0^1 \mathrm{d}y \frac{|v(z,y)|}{z+y}\rho(y) \biggr) \biggr|^2 \\ &\leqslant\mu^{-1}\int_0^1 \mathrm{d} x\biggl| \frac{1}{\pi^2}\int_0^1 \mathrm{d} z |a(x,z;\mu+\mathrm{i}\epsilon)-a(x,z;\mu)| \biggl[\int_0^1 \mathrm{d}y \frac{|v(z,y)|^2}{(z+y)^2}\biggr]^{1/2} \biggr|^2 \|\rho\|^2 \\ &\leqslant\mu^{-1}\int_0^1 \mathrm{d} x\biggl| \frac{1}{\pi^2}\int_0^1 \mathrm{d} z |a(x,z;\mu+\mathrm{i}\epsilon)\,{-}\,a(x,z;\mu)| C_0\biggl[\int_0^1 \mathrm{d}y \frac{(z^{b_*}{+}\,y^{b_*})^2}{(z+y)^2}\biggr]^{1/2} \biggr|^2 \|\rho\|^2. \end{aligned} \end{equation*} \notag $$
Using the estimate
$$ \begin{equation*} \biggl[\int_0^1 \mathrm{d}y \frac{(z^{b_*}+y^{b_*})^2}{(z+y)^2}\biggr]^{1/2}\leqslant \begin{cases} C\dfrac{z^{b_*}}{\sqrt{z}},&b_*<\dfrac12, \\ C |{\log z}|,& b_*=\dfrac12, \\ \mathrm{Const},& 1>b_*>\dfrac12, \end{cases} \end{equation*} \notag $$
we obtain
$$ \begin{equation} \|(\mathbf{B}(\mu+\mathrm{i}\epsilon)-\mathbf{B}(\mu))\|^2 \leqslant \frac{C}{\mu}\int_0^1 \mathrm{d} x \,G_\epsilon(x;\mu), \end{equation} \tag{4.9} $$
where
$$ \begin{equation*} G_\epsilon(x;\mu)= \biggl[\int_0^1 \mathrm{d} z\, |a(x,z;\mu+\mathrm{i}\epsilon)-a(x,z;\mu)| \frac{z^{b_*}}{\sqrt{z}} \biggr]^2 \end{equation*} \notag $$
(if $b_*<1/2$; the other cases are dealt with similarly). In order to pass to the limit on the right-hand side of (4.9), thereby verifying the limit relations in (4.8), we use the asymptotics
$$ \begin{equation*} a(x,y;\mu)= \frac{\sinh(\arccos(1/\mu)\log[x/y]/\pi)}{\sinh(\log[x/y]/2)} \frac{1}{\sqrt{\mu^2\!-\!1}(x\!+\!y)} + B_\mu(x,y)O\biggl(\frac{\log^{-m}(xy)}{\sqrt{xy}}\!\biggr) \end{equation*} \notag $$
for the resolvent kernel (see Lemma 7) for $\mu\in \omega_+\cup Q_+$, where $m$ is any natural number, the branch $\sqrt{\mu^2-1}$ is as defined above, $ B_\mu(x,y)$ is continuous with respect to $\mu$ on any compact set in $\omega_+\cup Q_+$ and bounded in $(x,y)$, and $\arccos(1/\mu)=-\mathrm{i}\operatorname{arcosh}(1/\mu)$ is bounded and imaginary, with $\operatorname{arcosh}(1/\mu)>0$ for $0<\mu<1$. The branch of $\arccos z$ is fixed by the condition $\arccos z>0$ for $ -1<z<1$ on the upper side of the cut, and the cut is made along $[-1,1]$. For $\mu\in \omega_+$ the first term in the asymptotic formula is the same and the second is of order $B_\mu(x,y)O((xy)^{\delta-1/2})$, $\delta>0$.

The above asymptotic formula implies that $G_\epsilon(x;\mu)$ has an integrable majorant for $\mu\in \omega_+\cup Q_+$:

$$ \begin{equation*} |G_\epsilon(x;\mu)|\leqslant C_\mu x^{2b_*-1}, \qquad b_*>0, \end{equation*} \notag $$
where $C_\mu$ is bounded in $\mu$ on any compact subset of $\omega_+\cup Q_+$. Letting $\epsilon\to 0$ on the right- and left-hand sides of (4.9) (with the use of Lebesgue’s dominated convergence theorem) and using the continuity of $a(x,y;\mu)$ in $\mu$ as $\mu\in \omega_+\cup Q_+$, we arrive at (4.8). Summarizing the above, we have the following result.

Lemma 4. Let $\mu\in\omega_+ \cup Q_+\setminus\mathcal N_+$, where $\mathcal N_+$ is a Lebesgue nullset. Then the operator $\mathbf{I}+\mathbf{B}(\mu) $ is boundedly invertible, and equation (4.7) has a unique solution $g_p$ in $ L_2([0,1])$ for any right-hand side in $L_2([0,1])$.

Recall that9 $\mathcal N_+=\bigl\{\mu\in \omega_+ \cup \overline{Q}_+\colon -1 \in \sigma(\mathbf{B}(\mu)) \bigr\}$.

The following result is a consequence of Lemmas 3 and 4.

Proposition 3. For all $\mu=\mu(p)\in \sigma_e(\mathbf{K})$ with the possible exception of a Lebesgue nullset, the generalized eigenfunction of the operator $\mathbf{K}$ has the representation

$$ \begin{equation*} \rho_p(x)=\mathcal P_p(x)+g_p(x), \qquad g_p\in L_2(0,1), \qquad p\in [0,\infty). \end{equation*} \notag $$

From Proposition 3 and the asymptotic formulae for Legendre functions we obtain an asymptotic estimate for the generalized eigenfunction:

$$ \begin{equation*} \rho_p(x)\sim \biggl( C(p)\biggl[\frac{x}{2}\biggr]^{-1/2-\mathrm{i}p} +C(-p)\biggl[\frac{x}{2}\biggr]^{-1/2+\mathrm{i}p}\biggr)(1+o(1)) \end{equation*} \notag $$
as $x\to 0$ for $p\geqslant 0$; $C(\,{\cdot}\,)$ can be written out explicitly. One may believe that the singular set $\mathcal N_+$ is empty in our setting — however, this does not follow from our approach to the proof of Proposition 3.

§ 5. Solutions of the functional-difference equation, $\Lambda\geqslant 1$

Using the results from the previous section we describe the asymptotic behaviour of the solutions to the functional-difference equation as $|\nu|\to\infty$, $\nu=|\nu|\mathrm{e}^{\mathrm{i}\psi}$, ${\psi\in(0,\pi/2]}$ for $\Lambda\geqslant 1$. The formula

$$ \begin{equation*} \rho_p(x)= \cos\pi\nu D(\nu)|_{x= {1}/{\cos\pi\nu}}\sqrt{w_0(x)}, \end{equation*} \notag $$
which relates $\rho_p(x)$ and $D(\nu)$ for all $\Lambda^{-1}=\mu(p)=1/\cos\pi p\in(0,1]$, $p\in [0,\infty)$, shows that $D(\nu)$ is the required meromorphic solution, which satisfies
$$ \begin{equation*} |D(\nu)|<\mathrm{Const}\biggl|\exp\biggl(-\frac{\pi|\nu|}2\biggr)\biggr|, \qquad |\nu|\to \infty, \end{equation*} \notag $$
on the imaginary axis and in the strip $\Pi_{1+\delta}$. However, this estimate is not sufficient for our purposes, so we obtain an asymptotic formula for $D$.

The set $C_e=\{\Lambda\colon \Lambda\geqslant 1\}$ will be referred to as the characteristic set, and the corresponding solutions of the functional-difference equation (2.5) are called (generalized) eigenfunctions of the essential characteristic set $C_e$. It is convenient to introduce the parameter $\tau$ and parameterize $\Lambda\geqslant 1$ by means of the relation

$$ \begin{equation*} \Lambda=:\sin\biggl(\frac{\pi}{2}+\mathrm{i}\tau\biggr)=\cosh\tau, \qquad \tau\geqslant 0. \end{equation*} \notag $$
(Note that we could as well take negative values of the parameter $\tau\leqslant 0$.)

The asymptotic behaviour of a meromorphic function is closely related to singularities of its Fourier transforms (in our setting, along the imaginary axis). Let us apply the Fourier transform along the imaginary axis,

$$ \begin{equation*} \chi(\zeta)=\int_{\mathrm{i}\mathrm{R}} \exp(\mathrm{i}\zeta\nu) h(\nu) \,\mathrm{d}\nu\quad\text{and} \quad h(\nu) =-\frac{\mathrm{v.p.}}{2\pi}\int_{\mathrm{i}\mathrm{R}} \exp(-\mathrm{i}\zeta\nu) \chi(\zeta)\,\mathrm{d}\zeta \end{equation*} \notag $$
to equation (2.5). After some simple algebra we obtain the relation
$$ \begin{equation*} \begin{gathered} \, \biggl[\sin\zeta-\sin\biggl(\frac{\pi}{2}+\mathrm{i}\tau\biggr)\biggr]{F}(\zeta) + \sin\biggl(\frac{\pi}{2}+\mathrm{i}\tau\biggr)\int_{\mathrm{i}\mathrm{R}} \exp(\mathrm{i}\zeta\nu) [{W}(\nu)+1]{D}(\nu) \,\mathrm{d}\nu=0, \\ F(\zeta)=\int_{\mathrm{i}\mathrm{R}} \exp(\mathrm{i}\zeta\nu) D(\nu) \,\mathrm{d}\nu, \end{gathered} \end{equation*} \notag $$
which we write in the form
$$ \begin{equation*} {F}(\zeta) = -\frac{\sin({\pi}/{2}+\mathrm{i}\tau)} {[\sin\zeta-\sin({\pi}/{2}+\mathrm{i}\tau)]} \int_{\mathrm{i}\mathrm{R}} \exp(\mathrm{i}\zeta\nu) [{W}(\nu)+1]{D}(\nu) \,\mathrm{d}\nu. \end{equation*} \notag $$

Let us verify that the integral on the right-hand side is holomorphic in some strip $\zeta\in\Pi_{\pi/2+q_*}= \{\zeta\in\mathbb C\colon -(\pi/2+q_*)<\operatorname{Re}\zeta<\pi/2+q_*\}$, $q_*>0$. This will show that the singularities which are closest to the imaginary axis (the poles of $F$) are at the points $\zeta=\pm ({\pi}/{2}+\mathrm{i}\tau)$ (note that $F$ is an even function). Assuming that $\operatorname{Re}\zeta>0$ we write the integral in the form

$$ \begin{equation*} \begin{aligned} \, \int_{\mathrm{i}\mathbb R}\exp(\mathrm{i}\zeta\nu) [{W}(\nu)+1]D(\nu) \,\mathrm{d}\nu &=\int_{\mathrm{i}\mathbb R} \exp(\mathrm{i}\zeta\nu) [W(\nu)+\mathrm{i}\operatorname{tg}(b\nu)]D(\nu) \,\mathrm{d}\nu \\ &\qquad+\int_{\mathrm{i}\mathbb R} \exp(\mathrm{i}\zeta\nu) [-\mathrm{i}\operatorname{tg}(b\nu)+\operatorname{sign}(\mathrm{i}\nu)]D(\nu) \,\mathrm{d}\nu \\ &\qquad +\int_{\mathrm{i}\mathbb R} \exp(\mathrm{i}\zeta\nu) [-\operatorname{sign}(\mathrm{i}\nu)+1]D(\nu) \,\mathrm{d}\nu. \end{aligned} \end{equation*} \notag $$
In the last term integration is actually along $\mathrm{i}\mathbb R_+$ since the factor $[-\operatorname{sign}(\mathrm{i}\nu)\,{+}\,1] $ vanishes on the negative part of the imaginary axis, hence the integral is holomorphic for $\operatorname{Re}(\zeta)\!>\!0$. Recalling the asymptotic formula ${W(\nu)\!=\!1\!+\!O(\exp\{\pm\mathrm{i}q_*\nu\})\!=\!\Phi}$ as $\nu\to \mathrm{i}\infty$ along the imaginary axis (in our case $q_*=2\min\{\Phi,\overline{\Phi}\}>0$), we find that the first integral is regular in the strip $\Pi(-{\pi}/{2}-q_*,{\pi}/{2}+q_*)$, where we assume that $b>q_*$. For sufficiently large $b>0$ the second integral is holomorphic in $\Pi(-[2b+{\pi}/{2}],2b+{\pi}/{2})$. Hence the function $F(\,{\cdot}\,) $ is holomorphic in $\Pi(-{\pi}/{2}-q_*,{\pi}/{2}+q_*)$, except at the poles at $\zeta=\pm ({\pi}/{2}+\mathrm{i}\tau)$, which are the closest ones to the imaginary axis. Using the inverse Fourier transform,
$$ \begin{equation*} D(\nu) =-\frac{\mathrm{v.p.}}{2\pi}\int_{\mathrm{i}\mathrm{R}} \exp(-\mathrm{i}\zeta\nu) F(\zeta)\,\mathrm{d}\zeta \end{equation*} \notag $$
and shifting the integration contour (note that $F$ decays rapidly in the strip along the imaginary axis) we evaluate the contribution of the residues at the poles closest to the imaginary axis. We have thus arrived at the following result.

Lemma 5. The asymptotic formula

$$ \begin{equation} D(\nu) =\frac{C_D }{\cos[\nu({\pi}/{2}+\mathrm{i}\tau)]} \bigl(1+O(\exp(-\delta_0|\nu|))\bigr) \end{equation} \tag{5.1} $$
holds for some $\delta_0>0$ as $|\nu|\to\infty$ in an arbitrary strip $\nu\in \Pi_q$ of finite width $2q$, where $C_D$ is a constant.

The correction to the leading term in the asymptotics (5.1) depends on the contribution of the poles of $F$ which are the next-to-nearest ones to the imaginary axis. Note that the solution $D(\,{\cdot}\,)$ is holomorphic outside the strip $|{\operatorname{Im}\nu}| < b$ for some positive $b$. On the other hand (5.1) holds in the strip $ \Pi_q$ of arbitrary (but fixed) width $2q$. From this observation we obtain the estimate $ |D(\nu)|<\mathrm{Const}|\exp(-|\nu|({\pi}/{2}) \sin|\psi|)|$ as $|\nu|\to\infty$, for all $\pm\psi\!\in\! [0,\pi/2]$, where $\nu\!=\!|\nu|\exp(\mathrm{i}\psi)$ and $|\nu|\to\infty$ outside the strip $|{\operatorname{Im} \nu}| <b$, and, in particular, on the contour $C_\psi^{b}$. In view of Lemma 2 and Proposition 3 this produces the following result.

Lemma 6. For each $\Lambda$ in the essential characteristic set $C_e=[1,\infty)$ there exists an even solution $D(\,{\cdot}\,)$ of the functional-difference equation (2.5) (a generalized eigenfunction of the essential characteristic set) in the class $\mathcal M$ and with asymptotic behaviour (5.1).

The next result follows from Lemma 6 and the properties of solutions of (2.5).

Proposition 4. The generalized eigenfunction of the essential spectrum (the case $E=-\gamma^2/(4\Lambda)\in [-\gamma^2/4,0)$) of the operator $A_s$ has the Watson-Bessel integral representation (2.1) in terms of the generalized eigenfunction $D\in \mathcal M$ of the essential characteristic set of the functional-difference equation (2.5), $\Lambda\in [1, \infty)$.

Note that to each point $E=-{\gamma^2}/({4\Lambda})\in [-\gamma^2/4,0)\subset \sigma_e(A_s)$ there corresponds a unique point in the essential characteristic set $\Lambda\in [1, \infty)$ of the functional-difference equation (2.5).

Now we investigate the asymptotic behaviour of the generalized eigenfunction with respect to the distance and discuss its physical interpretation from the point of view of wave scattering.

§ 6. Reduction to the Sommerfeld integral and asymptotics of the generalized eigenfunction of the operator $A_s$

The asymptotic behaviour of the generalized eigenfunction described by formulae (2.1) cannot be found using the asymptotic formulae as $r\to \infty$ from the Bessel functions under the integral sign, because this makes the integral diverge. This difficulty can be circumvented by using the Sommerfeld integral representation for the Bessel function in the form10

$$ \begin{equation*} J_\nu(\mathrm{i}\kappa r)= - \frac{1}{2\pi}\int_{\gamma_0^-}\mathrm{d}\alpha \exp(\kappa r\cos\alpha)\exp\biggl(\frac{\mathrm{i}\nu\pi}2-\mathrm{i}\alpha\nu\biggr), \end{equation*} \notag $$
where the contour $\gamma_0^-$ is as shown in Figure 4.

Substituting the Sommerfeld representation for Bessel functions into (2.1) and changing the order of integration (which is legitimate) we have

$$ \begin{equation} \begin{aligned} \, u_1(r,\varphi) &=\frac{1}{2\pi \mathrm{i} }\int_{\gamma_0^-}\mathrm{d}\alpha \exp(\kappa r\cos\alpha)2\biggl\{\frac{1}{2}\int_{C_0^b} \exp(-\mathrm{i}\alpha\nu) \frac{\sin(\nu\varphi)}{\sin(\nu\Phi)} H_1(\nu)\,\mathrm{d}\nu\biggr\} \\ &=\frac{1}{2\pi \mathrm{i} }\int_{\gamma_0^-}\mathrm{d}\alpha \exp(\kappa r\cos\alpha)2 F_1(\alpha, \varphi), \qquad \varphi\in [0,\Phi], \\ u_2(r,\varphi) &= \frac{1}{2\pi \mathrm{i} }\int_{\gamma_0^-}\mathrm{d}\alpha \exp(\kappa r\cos\alpha)2\biggl\{\frac{1}{2}\int_{C_0^b} \exp(-\mathrm{i}\alpha\nu) \frac{\sin(\nu\overline{\varphi} )}{\sin(\nu\overline{\Phi})} H_2(\nu)\,\mathrm{d}\nu\biggr\} \\ &=\frac{1}{2\pi \mathrm{i} }\int_{\gamma_0^-}\mathrm{d}\alpha \exp(\kappa r\cos\alpha)2 F_2(\alpha, \overline{\varphi} ), \qquad \varphi\in [\Phi,\pi], \end{aligned} \end{equation} \tag{6.1} $$
where $\operatorname{Im}\alpha <0$, $|{\operatorname{Re} \alpha}|<\pi+\delta_0$, $\delta_0>0$,
$$ \begin{equation} F_1(\alpha, \varphi)= \frac{1}{2}\int_{C_0^b} \exp(-\mathrm{i}\alpha\nu) \frac{\sin(\nu\varphi)}{\sin(\nu\Phi)} H_1(\nu)\,\mathrm{d}\nu \end{equation} \tag{6.2} $$
and
$$ \begin{equation} F_2(\alpha, \varphi)= \frac{1}{2}\int_{C_0^b} \exp(-\mathrm{i}\alpha\nu) \frac{\sin(\nu\overline{\varphi} )}{\sin(\nu\overline{\Phi})} H_2(\nu)\,\mathrm{d}\nu. \end{equation} \tag{6.3} $$
In the Sommerfeld representations (6.1) the functions $F_{1}(\cdot,\varphi)$, $F_{2}(\cdot,\overline{\varphi} )$ are extended to the half-strip $\alpha\in \Pi_{\pi}$, $\operatorname{Im}\alpha <0$. Let us use the fact that $H_{1,2}\in \mathcal M$, in particular, we will employ the estimate $ |H_{1,2}(\nu)|<\mathrm{Const}|\exp(-|\nu|({\pi}/{2}) \sin|\psi|)|$ as $|\nu|\to\infty$ (for all $\pm\psi\in [0,\pi/2]$, where $\nu= |\nu|\exp(\mathrm{i}\psi)$, that is, on the contour $C_\psi^b$). Using the deformation $C_0^b\to C_\psi^b\to C_{\pi/2}^b= (-\mathrm{i}\infty,\mathrm{i}\infty)$ of the integration contour in (6.2) and (6.3) we have
$$ \begin{equation} F_1(\alpha, \varphi)= \frac{1}{2 \mathrm{i}}\int_{C_{\pi/2}^b} \sin(\alpha\nu) \frac{\sin(\nu\varphi)}{\sin(\nu\Phi)} H_1(\nu)\,\mathrm{d}\nu \end{equation} \tag{6.4} $$
and
$$ \begin{equation} F_2(\alpha, \varphi)= \frac{1}{2\mathrm{i}}\int_{C_{\pi/2}^b}\sin(\alpha\nu) \frac{\sin(\nu\overline{\varphi} )}{\sin(\nu\overline{\Phi})} H_2(\nu)\,\mathrm{d}\nu, \end{equation} \tag{6.5} $$
where integration in (6.4) and (6.5) is along the imaginary axis, and the exponential is replaced by the sine function since the integrand is odd. Using the above estimates for $H_{1,2}$, we conclude that $F_1(\cdot,\varphi)$ and $F_2(\cdot,\varphi)$ are holomorphic in $\Pi_{\pi/2+[\Phi-\varphi]}$ and $\Pi_{\pi/2+[\overline{\Phi}-\overline{\varphi} ]}$, respectively. In addition, $F_1(\cdot,\varphi)$ and $F_2(\cdot,\varphi)$ can be extended meromorphically to the complex plane (we discuss this below). We can write (6.4) and (6.5) as
$$ \begin{equation} F_1(\alpha, \varphi) = \frac{1}{4 \mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty} \frac{\cos(\nu[\alpha\!-\!\varphi])\,{-}\cos(\nu[\alpha\!+\!\varphi])}{\sin(\nu\Phi)} H_1(\nu)\,\mathrm{d}\nu = f_1(\alpha\!+\!\varphi)\,{-}\,f_1(\alpha\!-\!\varphi) \end{equation} \tag{6.6} $$
and
$$ \begin{equation} F_2(\alpha, \varphi) = \frac{1}{4\mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty} \frac{\cos(\nu[\alpha\!-\!\varphi])\,{-}\cos(\nu[\alpha\!+\!\varphi])}{\sin(\nu\overline{\Phi})} H_2(\nu)\,\mathrm{d}\nu = f_2(\alpha\!+\!\overline{\varphi} )\,{-}\,f_2(\alpha\!-\!\overline{\varphi} ), \end{equation} \tag{6.7} $$
where
$$ \begin{equation} \begin{gathered} \, f_1(\alpha)= \frac{1}{4 \mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty} \frac{\cos(\nu\alpha)}{\sin(\nu\Phi)} H_1(\nu)\,\mathrm{d}\nu, \qquad \alpha\in\Pi_{\pi/2+\Phi}, \\ f_2(\alpha)= \frac{1}{4 \mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty} \frac{\cos(\nu\alpha)}{\sin(\nu\overline{\Phi})} H_2(\nu)\,\mathrm{d}\nu, \qquad \alpha\in\Pi_{\pi/2+\overline{\Phi}}. \end{gathered} \end{equation} \tag{6.8} $$
As a result, in view of (6.1) and (6.6)(6.8) the Sommerfeld integral representations assume the form
$$ \begin{equation} \begin{aligned} \, u_1(r,\varphi) &=\frac{1}{2\pi \mathrm{i} }\int_{\gamma_0}\mathrm{d}\alpha \exp(\kappa r\cos\alpha)[f_1(\alpha+\varphi)-f_1(\alpha-\varphi)] \\ &=\frac{1}{2\pi \mathrm{i} }\int_{\gamma_0}\mathrm{d}\alpha \exp(\kappa r\cos\alpha)2f_1(\alpha+\varphi), \qquad \varphi\in [0,\Phi], \\ u_2(r,\varphi) &=\frac{1}{2\pi \mathrm{i} }\int_{\gamma_0}\mathrm{d}\alpha \exp(\kappa r\cos\alpha)[f_2(\alpha+\overline{\varphi} )-f_2(\alpha-\overline{\varphi} )] \\ &= \frac{1}{2\pi \mathrm{i} }\int_{\gamma_0}\mathrm{d}\alpha \exp(\kappa r\cos\alpha)2f_2(\alpha+\overline{\varphi} ), \qquad \varphi\in [\Phi,\pi] \end{aligned} \end{equation} \tag{6.9} $$
(the contour $\gamma_0=\gamma_0^+\cup\gamma_0^-$ is as shown in Figure 4), and the Sommerfeld transformants $f_1$ and $f_2$ are even and holomorphic in the strips $\Pi_{\pi/2+\Phi}$ and $\Pi_{\pi/2+\overline{\Phi}}$, respectively.

6.1. The meromorphic extension of $f_1$ and $f_2$ and the Malyuzhinets’s functional equation

By construction the integral representations (6.9) satisfy equations (1.2); this can also be easily verified by direct substitution, because the integrals converge rapidly (and uniformly with respect to $r$ and $\varphi$). The Dirichlet boundary condition on the boundary half-plane is met. A substitution of the Sommerfeld integrals into the continuity boundary condition on $l$ transforms this condition into Malyuzhinets’s functional equation (see [11], § 3.4, Malyuzhinets’s theorem)

$$ \begin{equation} f_1(\alpha+\Phi)-f_1(\alpha-\Phi)= f_2(\alpha+\overline{\Phi})-f_2(\alpha-\overline{\Phi}). \end{equation} \tag{6.10} $$

Substituting the Sommerfeld representations into the Robin-type boundary condition (1.4) and integrating by parts we obtain

$$ \begin{equation*} \begin{aligned} \, &\frac{1}{\kappa r}\biggl(\frac{\partial u_1}{\partial \varphi} -\frac{\partial u_2}{\partial \varphi}\biggr) \bigg|_{\varphi=\Phi}- \frac{\gamma}{\kappa}u_1 (r,\Phi) \\ &\qquad =\frac{1}{2\mathrm{i}\pi}\int_{\gamma_0}\mathrm{d}\zeta \frac{\exp(\kappa r\cos\zeta)}{\kappa r} 2\biggl(-\sin\alpha[ f_1(\zeta+\Phi)+f_2(\zeta-\Phi)] \\ &\qquad\qquad -\frac{\gamma_1}{2\kappa}[ f_1(\zeta-\Phi)+f_2(\zeta-\overline{\Phi})]\biggr)=0. \end{aligned} \end{equation*} \notag $$
We set $\Lambda :={\gamma_1}/(2\kappa)=\sin\vartheta_\tau$, $\vartheta_\tau=\pi/2+\mathrm{i}\tau$. By the Sommerfeld’s integral inversion theorem (see § 3.4 in [11])
$$ \begin{equation} \begin{aligned} \, \notag &(\sin\alpha-\sin\vartheta_\tau)f_1(\alpha+\Phi) +(\sin\alpha-\sin\vartheta_\tau)f_2(\alpha+\overline{\Phi}) \\ &\qquad =(\sin\alpha+\sin\vartheta_\tau)f_1(-\alpha+\Phi) -(\sin\alpha+\sin\vartheta_\tau)f_2(-\alpha+\overline{\Phi}), \end{aligned} \end{equation} \tag{6.11} $$
where we have used the fact that $f_{1,2}$ is even. In an equivalent form, Malyuzhinets’s equations (6.10) and (6.11) read as
$$ \begin{equation} \begin{pmatrix} f_1(\alpha+\Phi) \\ f_2(\alpha+\overline{\Phi}) \end{pmatrix} = \begin{pmatrix} \dfrac{-\sin\vartheta_\tau}{\sin\alpha-\sin\vartheta_\tau} &\dfrac{-\sin\vartheta_\tau}{\sin\alpha-\sin\vartheta_\tau} \\ \dfrac{-\sin\alpha}{\sin\alpha-\sin\vartheta_\tau} &\dfrac{-\sin\alpha}{\sin\alpha-\sin\vartheta_\tau} \end{pmatrix} \begin{pmatrix} f_1(-\alpha+\Phi) \\ f_2(-\alpha+\overline{\Phi}) \end{pmatrix}. \end{equation} \tag{6.12} $$
The even functions $f_{1,2}$ are holomorphic in the strips $\Pi_{\pi/2+\Phi}$ and $\Pi_{\pi/2+\overline{\Phi}}$, respectively; they extend as meromorphic functions to the right half-plane (and, as they are even, to the left half-plane) via Malyuzhinets’s functional equations (6.12). Indeed, if the arguments of $f_{1,2}$ on the right-hand side of (6.12) range in the strip of holomorphy, then on the left-hand side the argument ranges in this strip shifted right. For example, for the first equation
$$ \begin{equation*} f_1(\alpha+2\Phi)=\frac{-\sin\vartheta_\tau}{\sin(\alpha+\Phi)-\sin\vartheta_\tau}(f_1(-\alpha)+ f_2(-\alpha+\overline{\Phi}-\Phi)), \end{equation*} \notag $$
if on the right-hand side we assume that $-\alpha\in \Pi_{\pi/2+\Phi}$ in the argument of $f_1(-\alpha)$, then the argument of $f_2$ lies in the strip $-(\pi/2+\overline{\Phi})<\operatorname{Re} [\alpha-\overline{\Phi}+\Phi]< \pi/2+\overline{\Phi}$ (or, what is the same, $-(\pi/2+\Phi)<\operatorname{Re} \alpha< \pi/2+2\overline{\Phi}-\Phi$). In addition, the argument of $f_1(\zeta)$, where $\zeta=\alpha+2\Phi$, varies in the strip $-\pi/2+\Phi<\operatorname{Re}\zeta < \pi/2+3\Phi$. This gives us the meromorphic extension of $f_1(\,{\cdot}\,)$ to the strip $-\pi/2+\Phi<\operatorname{Re}\zeta < \pi/2+3\Phi$. A similar situation also occurs with the second equation and with the extension of $f_2$ to the right of the strip of holomorphy $\Pi_{\pi/2+\overline{\Phi}}$. Analysing singularities of the factor on the right-hand side, we find that the pole of $f_1(\,{\cdot}\,)$ closest to the imaginary axis is at $\alpha =\Phi+\vartheta_\tau$, $\vartheta_\tau=\pi/2+\mathrm{i}\tau$; this pole lies on the boundary of the strip of holomorphy, and, in addition, in a neighbourhood of the pole
$$ \begin{equation*} f_1(\alpha)=\frac{A_1}{\alpha -[\Phi+\pi/2+\mathrm{i}\tau]}+\dotsb. \end{equation*} \notag $$
From the second equation in (6.12) we obtain
$$ \begin{equation*} f_2(\alpha)=\frac{A_2}{\alpha -[\overline{\Phi}+\pi/2+\mathrm{i}\tau]}+\dotsb \end{equation*} \notag $$
near the pole $\alpha =\overline{\Phi}+\vartheta_\tau$, which is the closest one to the imaginary axis. The shifts in Malyuzhinets’s equations are parallel to the real axis, hence the poles of the meromorphic functions which are next-to-nearest ones to the imaginary axis lie to the right of the strips of holomorphy, on the line $\operatorname{Im} \alpha= \tau$, so that, by parity arguments, the poles to the left of the strip of holomorphy are symmetric to them about the origin.

Remark 2. From the integral expressions (6.8) for $f_{1,2}$ it follows that in the strips of holomorphy we have

$$ \begin{equation} f_{1,2}(\alpha)\to \mathrm{C}_{1,2}, \qquad \operatorname{Im}\alpha\to \infty. \end{equation} \tag{6.13} $$
We also note that $u_{1,2}(0,\varphi)=0$.

Proposition 5. The generalized eigenfunction $u=u_j$ in $\Omega_j$, $j=1,2$, of the essential spectrum $E=-{\gamma^2}/({4\Lambda})\in [-\gamma^2/4,0)$ of the operator $A_s$ satisfies the Sommerfeld integral representation (6.9), where the Sommerfeld transformants $f_{1,2}$ are meromorphic even functions holomorphic in the strips $\Pi_{\pi/2+\Phi}$ and $\Pi_{\pi/2+\overline{\Phi}}$, respectively. They are related by the transformation (6.8) to the generalized eigenfunction $H_{1,2}\in \mathcal M$ of the essential characteristic set of the functional-difference equation (2.5) (for $\Lambda\geqslant 1$).

6.2. The asymptotic behaviour of eigenfunctions and physical interpretation

Now we find the asymptotics (as $r\to\infty$) of the generalized eigenfunction $u$ defined by the Sommerfeld integral (6.9). Unlike the Watson-Bessel integral representation, the Sommerfeld integrals are well suited for finding the asymptotics with respect to distance. Indeed, let us apply the saddle-point method to the integrals (6.9). Note that $\cos\alpha$ in the exponent has stationary points $\pm \pi$, which are roots of the equation $\sin\alpha=0$. To estimate the integral asymptotically we deform the integration contours $\gamma_0=\gamma_0^+\cup\gamma_0^-$ into the stationary ones $\gamma_0^\pi\cup\gamma_0^{-\pi}$ (see Figure 4). The latter contours pass through the points $\pm \pi$, respectively, and at these points they are parallel to the imaginary axis. In this deformation the poles of the integrand cross; in particular, for $f_1$ the main contribution comes from the poles $\alpha+\varphi =\pm[\Phi+\pi/2+\mathrm{i}\tau]$, $\tau>0$, which are the closest ones to the imaginary axis. Note that these poles migrate with varying $\varphi\in[0,\Phi]$. For $\varphi=\Phi$ the pole $\alpha=\pi/2+\mathrm{i}\tau$ is closest to the imaginary axis and makes the main contribution to the asymptotic formula, whereas the contributions from the remaining poles, and, of course, from the saddle-points $I_{\pm \pi}=O(\mathrm{e}^{-\kappa r}/\sqrt{\kappa r})$, are smaller in order. As a result, we obtain

$$ \begin{equation} \begin{aligned} \, &u_1(r,\varphi) =2A_1\exp\biggl(\kappa r\cos\biggl[\Phi-\varphi+\frac\pi2+\mathrm{i}\tau\biggr]\biggr) \\ &\quad +\sum_{|{\operatorname{Re} \alpha_p^1}|<\pi}\!\!\!\!\!\exp\bigl(\kappa r\cos[\alpha_p^1-\varphi]\bigr) 2\operatorname*{res}_{\alpha_p^1}f_1(\alpha+\varphi)+ O\biggl(\frac{\mathrm{e}^{-\kappa r}}{\sqrt{\kappa r}}\biggr), \quad \varphi\in [0,\Phi], \\ &u_2(r,\varphi) =2A_2\exp\biggl(\kappa r\cos\biggl[\overline{\Phi}-\overline{\varphi} +\frac\pi2+\mathrm{i}\tau\biggr]\biggr) \\ &\quad +\sum_{|{\operatorname{Re} \alpha_p^2}|<\pi} \!\!\!\!\!\exp\bigl(\kappa r\cos[\alpha_p^2-\overline{\varphi} ]\bigr) 2\operatorname*{res}_{\alpha_p^2}f_2(\alpha+\overline{\varphi} )+ O\biggl(\frac{\mathrm{e}^{-\kappa r}}{\sqrt{\kappa r}}\biggr), \quad \varphi\in [\Phi,\pi], \end{aligned} \end{equation} \tag{6.14} $$
where the contribution from the next-to-leading poles (that is, the ones with the main contribution) is represented as the sum of the residues at the poles that, along with the leading ones, are captured under the deformation of the contours to stationary ones.

A straightforward analysis of the leading terms of the asymptotic formulae (6.14) shows that the expression

$$ \begin{equation} \begin{aligned} \, \notag u_1 \sim u_1^{\mathrm{out}} &:= 2A_1 \exp\biggl(\kappa r\cos\biggl[\Phi-\varphi+\frac\pi2+\mathrm{i}\tau)\biggr]\biggr) \\ &\,= 2A_1\exp\bigl(-\kappa r\{\sin[\Phi-\varphi]\cosh(\tau)-\mathrm{i}\cos[\Phi-\varphi]\sinh\tau\}\bigr) \end{aligned} \end{equation} \tag{6.15} $$
(for $\tau>0)$ is bounded on $l$ (that is, on the support of the potential) and decays exponentially as $\kappa r\to\infty$ in the remaining directions in $\Omega_1$. The asymptotic behaviour of $u_2$ in $\Omega_2$ is similar:
$$ \begin{equation} \begin{aligned} \, \notag u_2\sim u_2^{\mathrm{out}} &:= 2A_2\exp\biggl(\kappa r\cos\biggl[\overline{\Phi}-\overline{\varphi} +\frac\pi2+\mathrm{i}\tau\biggr]\biggr) \\ &\,= 2A_2\exp\bigl(-\kappa r\{\sin[\overline{\Phi}-\overline{\varphi} ]\cosh(\tau)-\mathrm{i}\cos[\overline{\Phi}-\overline{\varphi} ]\sinh\tau\}\bigr). \end{aligned} \end{equation} \tag{6.16} $$
In diffraction theory the expressions (6.15) and (6.16) are interpreted as a surface wave traveling outwards towards infinity from the origin along the ray $l$, bounded on $l$, and decreasing exponentially as $\kappa r\to\infty$ in the other directions (for details, see [11], § 6.4.1). So, for $\tau>0$ the above generalized eigenfunction $u$ is interpreted as a surface wave $u^{\mathrm{out}}$ traveling outwards towards infinity.

So far, we have been concerned with the case $\tau>0$. The case $\tau<0$ is dealt with quite similarly. In this case $\Lambda=\sin\vartheta_\tau=\sin(\pi/2+\mathrm{i}\tau)\geqslant 1$. The expressions (6.14) for the generalized eigenfunction have the same form also in this case. However, now the asymptotic (and therefore physical) interpretation of the generalized eigenfunction is different. For $\tau<0$ the above generalized eigenfunction $u$ is interpreted as a surface wave $u^{\mathrm{in}} $ incoming from infinity. So there are two types of generalized eigenfunctions, which correspond to outgoing ($\tau>0$) and incoming ($\tau<0$) surface waves and are exponentially localized near the support $l$ of the singular potential.

Proposition 6. For almost all $E\in [-\gamma^2/4,0)$ there exist generalized eigenfunctions $u=u_j$ in $\Omega_j$, $j=1,2$, of two types corresponding to outgoing ($\tau>0$, $u\sim u^{\mathrm{out}}$) and incoming ($\tau<0$, $u\sim u^{\mathrm{in}}$) surface waves. The generalized eigenfunctions have Sommerfeld integral representation (6.9), correspond to the essential spectrum of the operator $A_s$ and satisfy the asymptotic representation (6.14).

Problems of scattering of a surface wave incoming from infinity along a side of an infinite polygon with Robin-type conditions on the boundary were considered in [17]. The main result of [17] is asymptotic formulae for scattered waves. A similar statement of the scattering problem and the corresponding results can also be obtained for the generalized Jost problem considered in this paper (see Figure 1). The answers can be obtained in terms of ‘outgoing and incoming’ eigenfunctions, which were constructed above.

§ 7. Appendix. Asymptotics of $a(x,y;\mu)$ as $(x,y)\to 0$

Let us use the expression (4.6) for the resolvent kernel and the following representation for the Legendre functions (see [14], 8.772(1)):

$$ \begin{equation*} \begin{aligned} \, P_{\mathrm{i}p-1/2}\biggl(\frac 1x\biggr) &=\frac{\Gamma(\mathrm{i}p)}{\sqrt{\pi}\,\Gamma(\mathrm{i}p+1/2)} \biggl[\frac{x}{2}\biggr]^{-\mathrm{i}p+1/2}F\biggl(\frac{3/2-\mathrm{i}p}{2},\frac{1/2-\mathrm{i}p}{2}, 1-\mathrm{i}p; x^2\biggr) \\ &\qquad+ \frac{\Gamma(-\mathrm{i}p)}{\sqrt{\pi}\,\Gamma(-\mathrm{i}p+1/2)} \biggl[\frac{x}{2}\biggr]^{\mathrm{i}p+1/2}F\biggl(\frac{3/2+\mathrm{i}p}{2},\frac{1/2+\mathrm{i}p}{2}, 1+\mathrm{i}p; x^2\biggr), \end{aligned} \end{equation*} \notag $$
where $0<x<1$, $p\geqslant 0$ and $F$ is the hypergeometric function:
$$ \begin{equation*} F(\alpha,\beta,\gamma;t)= \frac{\Gamma(\gamma)}{\Gamma(\alpha)\Gamma(\beta)}\sum_{n\geqslant 0} \frac{\Gamma(\alpha+n)\Gamma(\beta+n)}{\Gamma(\gamma+n)}\,\frac{t^n}{n!}, \qquad |t|<1. \end{equation*} \notag $$

For further transformations and the evaluation of the asymptotics it is convenient to write

$$ \begin{equation*} \mathrm{P}(x,p):= \frac{\Gamma(\mathrm{i}p)}{\sqrt{\pi}\,\Gamma(\mathrm{i}p+1/2)} \biggl[\frac{x}{2}\biggr]^{-\mathrm{i}p+1/2} \end{equation*} \notag $$
and
$$ \begin{equation*} \mathrm{D}(x,p):=F\biggl(\frac{3/2-\mathrm{i}p}{2},\frac{1/2-\mathrm{i}p}{2}, 1-\mathrm{i}p; x^2\biggr), \end{equation*} \notag $$
where we use the fact that $\mathrm{D}(x,p)= 1 +O(x^2)$ as $x\to 0$. Using this notation we write $a(x,y;\mu)$ as
$$ \begin{equation} \begin{aligned} \, \notag &a(x,y;\mu) =\frac{1}{2 xy}\int_{-\infty}^\infty\mathrm{d}p\, \frac{\pi p\tanh(\pi p)}{ \mu\cosh(\pi p)-1} \bigl\{\bigl(\mathrm{P}(x,-p)+\mathrm{P}(x,p)\bigr) \bigl(\mathrm{P}(y,p)+\mathrm{P}(y,-p)\bigr) \\ \notag &\quad +\bigl(\mathrm{P}(x,-p)+\mathrm{P}(x,p)\bigr)\bigl(\mathrm{P}(y,p) [\mathrm{D}(y,p)-1]+ \mathrm{P}(y,-p) [\mathrm{D}(y,-p)-1]\bigr) \\ &\quad + \bigl(\mathrm{P}(y,-p)+\mathrm{P}(y,p)\bigr)\bigl(\mathrm{P}(x,p) [\mathrm{D}(x,p)-1]+ \mathrm{P}(x,-p) [\mathrm{D}(x,-p)-1]\bigr) \bigr\}. \end{aligned} \end{equation} \tag{7.1} $$
In the first line of (7.1) we expand brackets and make the change $-p\to p$ in the integral involving $\mathrm{P}(x,-p)\mathrm{P}(y,p)$. As a result, we obtain the integral containing $2\mathrm{P}(x,p)\mathrm{P}(y,-p)$. Proceeding similarly with the product $\mathrm{P}(x,-p)\mathrm{P}(y,-p)$ we find that11
$$ \begin{equation} \begin{aligned} \, \notag &a(x,y;\mu) =\frac{1}{2 xy}\int_{-\infty}^\infty\mathrm{d}p \frac{\pi p\tanh(\pi p)}{ \mu\cosh(\pi p)-1} \bigl\{ 2\mathrm{P}(x,p)\mathrm{P}(y,-p)+ 2\mathrm{P}(x,p)\mathrm{P}(y,p) \\ \notag &\qquad +\bigl(\mathrm{P}(x,-p)+\mathrm{P}(x,p)\bigr)\bigl(\mathrm{P}(y,p)[\mathrm{D}(y,p)-1]+ \mathrm{P}(y,-p) [\mathrm{D}(y,-p)-1]\bigr) \\ &\qquad +\bigl(\mathrm{P}(y,-p)+\mathrm{P}(y,p)\bigr)\bigl(\mathrm{P}(x,p) [\mathrm{D}(x,p)-1]+ \mathrm{P}(x,-p) [\mathrm{D}(x,-p)-1]\bigr) \bigr\}. \end{aligned} \end{equation} \tag{7.2} $$
The last two lines in (7.2) contain the factors $[\mathrm{D}(x,\pm p)-1] =O(x^2)$ and ${[\mathrm{D}(y,\pm p)-1] =O(y^2)}$ is square brackets, and so it is natural no assume that, after integration, they will be small relative to the integral on the first line. We set
$$ \begin{equation*} \begin{aligned} \, R_+(x,y;\mu) &:= \frac{1}{2 xy}\int_{-\infty}^\infty\mathrm{d}p\, \frac{\pi p\tanh(\pi p)}{ \mu\cosh(\pi p)-1} \mathrm{P}(x,p)\mathrm{P}(y,-p) \\ &=\frac{1}{4\sqrt{xy}}\int_{-\infty}^\infty\mathrm{d}p\, \frac{ p\tanh(\pi p)}{\mu\cosh(\pi p)\!-\!1} \frac{\Gamma(\mathrm{i}p)\Gamma(-\mathrm{i}p)}{\Gamma(\mathrm{i}p\!+\!1/2)\Gamma(-\mathrm{i}p\!+\!1/2)} \exp\biggl(-\mathrm{i}p\log\frac xy\biggr) \end{aligned} \end{equation*} \notag $$
and
$$ \begin{equation*} \begin{aligned} \, R_-(x,y;\mu) &:= \frac{1}{2 xy}\int_{-\infty}^\infty\mathrm{d}p\, \frac{\pi p\tanh(\pi p)}{ \mu\cosh(\pi p)-1} \mathrm{P}(x,p)\mathrm{P}(y,p) \\ &=\frac{1}{4\sqrt{xy}}\int_{-\infty}^\infty\mathrm{d}p\, \frac{ p\tanh(\pi p)}{ \mu\cosh(\pi p)-1} \frac{\Gamma^2(\mathrm{i}p)}{\Gamma^2(\mathrm{i}p+1/2)} \exp\biggl(-\mathrm{i}p\log\frac{xy}4\biggr). \end{aligned} \end{equation*} \notag $$
We also define $R(x,y;\mu)$ by the equality
$$ \begin{equation*} a(x,y;\mu)=2R_+(x,y;\mu)+2R_-(x,y;\mu)+R(x,y;\mu), \end{equation*} \notag $$
which follows from (7.2).

Let us now obtain an asymptotic estimate for $R_+(x,y;\mu)$. To this end we use the relations

$$ \begin{equation*} \Gamma(\mathrm{i}p)\Gamma(-\mathrm{i}p)=\frac{\pi}{p\sinh\pi p}\quad\text{and} \quad \Gamma\biggl(\mathrm{i}p+\frac12\biggr)\Gamma\biggl(-\mathrm{i}p+\frac12\biggr)=\frac{\pi}{\cosh\pi p} \end{equation*} \notag $$
to simplify the integrand. Since the integrand is even, we find that
$$ \begin{equation*} R_+(x,y;\mu)= \frac{1}{2\sqrt{xy}}\int_{0}^\infty\mathrm{d}p \, \frac{\cos(p\log(x/y))}{ \mu\cosh(\pi p)-1}. \end{equation*} \notag $$
It is noteworthy that the integral on the right-hand side can be evaluated explicitly using formula 3.983(1) in [14] and analytic continuation with respect to the parameters (provided that $\mu>1$ and $x/y>0$). Namely, we have
$$ \begin{equation*} \begin{aligned} \, R_+(x,y;\mu) &=\frac{1}{2\sqrt{xy}}\, \frac{\sinh(\pi^{-1}\arccos(\mu^{-1})\log(x/y))}{\sqrt{\mu^2-1}\sinh\log(x/y)} \\ &= \frac{1}{x+y}\, \frac{\sinh(\pi^{-1}\arccos(\mu^{-1})\log(x/y))} {2\sqrt{\mu^2-1}\sinh(\log(x/y)/2)}. \end{aligned} \end{equation*} \notag $$
The branches of multi-valued functions are fixed as follows. For $\arccos(\,{\cdot}\,)$ the cut is along the interval $(-\infty,1]\cup[1\infty)$, and we assume that $\arccos z$ is positive for $z\in (0,1)$. Note that the extension for $z=1/\mu>1$ and also for $\mu \in \omega_+\cup Q_+$ (that is, to the rectangle) is unique. The branch of $\sqrt{\mu^2-1}$ is defined similarly: the cut is along $[-1,1]$, and we assume that $\sqrt{\mu^2-1}>0$ for $\mu>1$. Note that the limit of $R_+(x,y;1)$ as $\mu\to 1$ exists (see also [5], § 4.1). In addition, $R_+(x,y;\mu)$ depends continuously on $\mu\in \omega_+\cup Q_+$.

We estimate $R_-(x,y;\mu)$. We need the following formula ($\mu\in\omega_+$):

$$ \begin{equation*} R_-(x,y;\mu):= \frac{1}{4\sqrt{xy}}\int_{-\infty}^\infty \frac{\mathrm{d}p }{ \mu\cosh(\pi p)\!-\!1} \frac{\Gamma(\mathrm{i}p)}{\Gamma(\mathrm{i}p\!+\!1/2)} \frac{\Gamma(-\mathrm{i}p\!+\!1/2)}{\Gamma(-\mathrm{i}p)} \exp\biggl(-\mathrm{i}p\log\frac{xy}4\biggr). \end{equation*} \notag $$
It is clear that $\operatorname{Re}(\mathrm{i}p\log(xy/4))>0$ for $\operatorname{Im} p >0$. Using this observation we deform the integration contour along $\mathbb R$ into the contour $\mathbb R+\mathrm{i} h$, $h>0$. Note that in the process the poles of the integrand at the zeros of $\mu\cosh(\pi p)-1$ cross with the poles of ${\Gamma(\mathrm{i}p)}{\Gamma(-\mathrm{i}p+1/2)}$. If $\operatorname{Im} (\frac{1}{\pi}\operatorname{arcosh}\mu^{-1}) <h < 1$, then by the residue theorem the main contribution is made by the pole
$$ \begin{equation*} p(\mu):=\frac{1}{\pi}\log\biggl(\frac{1-\mathrm{i}\sqrt{\mu^2-1}}{\mu}\biggr), \end{equation*} \notag $$
where the branch is as above. The other roots of the equation $\cosh(\pi p)-1/\mu=0$ in the upper half-plane are as follows: $ p(\mu) +2 \mathrm{i}m$, $m=0,1, 2,\dots$ . As a result,
$$ \begin{equation*} R_-(x,y;\mu):= O\bigl([xy]^{\delta_*(\mu)-1/2}\bigr), \end{equation*} \notag $$
where $\delta_*(\mu):=\operatorname{Im} p(\mu)>0$, $\mu\in\omega_+$. However, if $\mu\in Q_+$ (that is, $\mu$ is positive), then the equation has two roots $\pm p(\mu)$ on the real axis. Here the integration contour is deformed into the contour $\mathcal L_\mu$ which, in the expression for $R_-$, goes below the poles $p(\mu)$ and above the poles $-p(\mu)$. So we have
$$ \begin{equation*} R_-(x,y;\mu):= \frac{1}{4\sqrt{xy}}\int_{\mathcal L_\mu}\frac{\mathrm{d}p }{ \mu\cosh(\pi p)-1} \frac{\Gamma(\mathrm{i}p)}{\Gamma(\mathrm{i}p\!+\!1/2)} \frac{\Gamma(-\mathrm{i}p\!+\!1/2)}{\Gamma(-\mathrm{i}p)} \exp\biggl(-\mathrm{i}p\log\frac{xy}4\biggr). \end{equation*} \notag $$
Integrating by parts in the last integral we obtain the estimate
$$ \begin{equation*} \begin{aligned} \, &R_-(x,y;\mu) = \frac{(-1)^{m}}{4\sqrt{xy}}\,\frac{1}{(-\mathrm{i}\log(xy/4))^m} \\ &\qquad \times\int_{\mathcal L_\mu}\mathrm{d}p\, \exp\biggl(-\mathrm{i}p\log\frac{xy}4\biggr) \frac{\mathrm{d}^m}{\mathrm{d} p^m}\biggl\{\frac{1}{\mu\cosh(\pi p)-1}\,\frac{\Gamma(\mathrm{i}p)}{\Gamma(\mathrm{i}p+1/2)}\, \frac{\Gamma(-\mathrm{i}p+1/2)}{\Gamma(-\mathrm{i}p)}\biggr\} \end{aligned} \end{equation*} \notag $$
for each $m=1,2,\dots$, $\mu\in Q_+$. Finally, $R(x,y;\mu)$ is estimated by $x^2R_-(x,y;\mu)$ or $y^2R_-(x,y;\mu)$ as $(x,y)\to(0,0)$ and $\mu\in \omega_+\cup Q_+$.

Lemma 7. The following asymptotic formula holds:

$$ \begin{equation*} a(x,y;\mu)= \frac{1}{x+y}\, \frac{\sinh(\pi^{-1}\arccos(\mu^{-1})\log(x/y))} {\sqrt{\mu^2-1}\sinh({\log(x/y)}/{2})}+\widetilde R_-(x,y;\mu), \end{equation*} \notag $$
where
$$ \begin{equation*} \widetilde R_-(x,y;\mu)= B_\mu(x,y) O\biggl(\frac{|{\log^{-m}(xy)}|}{\sqrt{xy}}\biggr), \qquad (x,y)\to(0,0), \end{equation*} \notag $$
$m=1, 2,\dots$ is arbitrary, and
$$ \begin{equation*} B_\mu(x,y) = \int_{\mathcal L_\mu}\mathrm{d}p \exp\biggl(-\mathrm{i}p\log\frac{xy}4\biggr) \frac{\mathrm{d}^m}{\mathrm{d} p^m}\biggl\{\frac{1}{ \mu\cosh(\pi p)\!-\!1}\,\frac{\Gamma(\mathrm{i}p)}{\Gamma(\mathrm{i}p\!+\!1/2)}\, \frac{\Gamma(-\mathrm{i}p\!+\!1/2)}{\Gamma(-\mathrm{i}p)}\biggr\} \end{equation*} \notag $$
is bounded in $(x,y)$. The above estimate is uniform with respect to the parameter $\mu$ on an arbitrary compact subset of $ \omega_+\cup Q_+$.


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Citation: M. A. Lyalinov, “On eigenfunctions of the essential spectrum of the model problem for the Schrödinger operator with singular potential”, Sb. Math., 214:10 (2023), 1415–1441
Citation in format AMSBIB
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\paper On eigenfunctions of the essential~spectrum of the model problem for the Schr\"odinger operator with singular potential
\jour Sb. Math.
\yr 2023
\vol 214
\issue 10
\pages 1415--1441
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