We study the existence of localized waves that can propagate in an acoustic medium bounded by two thin semi-infinite elastic membranes along their common edge. The membranes terminate an infinite ...wedge that is filled by the medium, and are rigidly connected at the points of their common edge. The acoustic pressure of the medium in the wedge satisfies the Helmholtz equation and the third-order boundary conditions on the bounding membranes as well as the other appropriate conditions like contact conditions at the edge. The existence of such localized waves is equivalent to existence of the discrete spectrum of a semi-bounded self-adjoint operator attributed to this problem. In order to compute the eigenvalues and eigenfunctions, we make use of an integral representation (of the Sommerfeld type) for the solutions and reduce the problem to functional equations. Their nontrivial solutions from a relevant class of functions exist only for some values of the spectral parameter. The asymptotics of the solutions (eigenfunctions) is also addressed. The far-zone asymptotics contains exponentially vanishing terms. The corresponding solutions exist only for some specific range of physical and geometrical parameters of the problem at hand.
DOI 10.1134/S1061920823030068
In this paper, the long-time asymptotics of the solution to the Cauchy problem is described by means of the evolution unitary group of the self-adjoint Mehler operator. Spectral analysis of the ...latter operator is also discussed.
In the present paper, eigenfunctions of essential and discrete spectrum are constructed. Integral representations and asymptotics of the eigenfunctions at far distances are obtained.
This work studies functional difference equations of the second order with a potential belonging to a special class of meromorphic functions. The equations depend on a spectral parameter. ...Consideration of this type of equations is motivated by applications in diffraction theory and by construction of eigenfunctions for the Laplace operator in angular domains. In particular, such eigenfunctions describe eigenoscillations of acoustic waves in angular domains with ‘semitransparent’ boundary conditions. For negative values of the spectral parameter, we study essential and discrete spectrum of the equations and describe properties of the corresponding solutions. The study is based on the reduction of the functional difference equations to integral equations with a symmetric kernel. A sufficient condition is formulated for the potential that ensures existence of the discrete spectrum. The obtained results are applied for studying the behaviour of eigenfunctions for the Laplace operator in adjacent angular domains with the Robin-type boundary conditions on their common boundary. At infinity, the eigenfunctions vanish exponentially as was expected. However, the rate of such decay depends on the observation direction. In particular, in a vicinity of some directions, the regime of decay is switched from one to another and such asymptotic behaviour is described by a Fresnel-type integral.
This work deals with the spectral properties of the functional-difference equations that arise in a number of applications in the diffraction of waves and quantum scattering. Their link with some of ...the spectral properties of perturbations of the Mehler operator is addressed. The latter naturally arise in studies of functional-difference equations of the second order with a meromorphic potential which depend on a characteristic parameter. In particular, this kind of equations is frequently encountered with in the asymptotic treatment of eigenfunctions of the Robin Laplacians in wedge- or cone-shaped domains. The unperturbed selfadjoint Mehler operator is studied by means of the modified Mehler–Fock transform. Its resolvent and spectral measure are described. These results are obtained by use of some additional analysis applied to the known Mehler formulas. For a class of compact perturbations of this operator, sufficient conditions of existence and finiteness of the discrete spectrum are then discussed. Applications to the functional-difference equations are also addressed. An example of a problem leading to the study of the spectral properties for a functional-difference equation is considered. The corresponding eigenfunctions and characteristic values are found explicitly in this case.
The eigenvalues and eigenfunctions of the discrete spectrum for Robin Laplacians in an angle are constructively computed by means of the Sommerfeld integral and of the Malyuzhinets functional ...equations.
We study the asymptotics with respect to distance for the eigenfunction of the Schrödinger operator in a half-plane with a singular
-potential supported by two half-lines. Such an operator occurs in ...problems of scattering of three one-dimensional quantum particles with point-like pair interaction under some additional restrictions, as well as in problems of wave diffraction in wedge-shaped and cone-shaped domains. Using the Kontorovich–Lebedev representation, the problem of constructing an eigenfunction of an operator reduces to studying a system of homogeneous functional-difference equations with a characteristic (spectral) parameter. We study the properties of solutions of such a system of second-order homogeneous functional-difference equations with a potential from a special class. Depending on the values of the characteristic parameter in the equations, we describe their nontrivial solutions, the eigenfunctions of the equation. The study of these solutions is based on reducing the system to integral equations with a bounded self-adjoint operator, which is a completely continuous perturbation of the matrix Mehler operator. For a perturbed Mehler operator, sufficient conditions are proposed for the existence of a discrete spectrum to the right of the essential spectrum. Conditions for the finiteness of the discrete spectrum are studied. These results are used in the considered problem in the half-plane. The transformation from the Kontorovich–Lebedev representation to the Sommerfeld integral representation is used to construct the asymptotics with respect to the distance for the eigenfunction of the Schrödinger operator under consideration.
A formal approach for the construction of the Green’s function in a polygonal domain with the Dirichlet boundary conditions is proposed. The complex form of the Kontorovich–Lebedev transform and the ...reduction to a system of integral equations is employed. The far-field asymptotics of the wave field is discussed.
The two-dimensional (2D) domain under study is bounded from below by two semi-infinite and, between them, two finite straight lines; on each of the straight lines (segments), a usually individual ...impedance boundary condition is imposed. An incident surface wave, propagating from infinity along one semi-infinite segment of the polygonal domain, excites outgoing surface waves both on the same segment (a reflected wave) and on the second semi-infinite segment (a transmitted wave); in addition, a circular (cylindrical) outgoing wave will be generated in the far field. The scattered wave field satisfies the Helmholtz equation and the Robin (in other words, impedance) boundary conditions as well as some special integral form of the Sommerfeld radiation conditions. It is shown that a classical solution of the problem is unique. By the use of some known extension of the Sommerfeld–Malyuzhinets technique, the problem is reduced to functional Malyuzhinets equations and then to a system of integral equations of the second kind with integral operator depending on a characteristic parameter. The Fredholm property of the equations is established, which also leads to the existence of the solution for noncharacteristic values of the parameter. From the Sommerfeld integral representation of the solution, the far-field asymptotics is developed. Numerical results for the scattering diagram are also presented.
