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 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 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.
An acoustic incident plane wave completely illuminates a narrow convex cone satisfying the impedance boundary condition on its surface. The wave field at far distances from the vertex of the cone and ...in some close neighborhood of the cone’s surface is asymptotically computed.
The paper deals with the asymptotic description of a diffraction pattern similar to the classical Weyl–Van der Pol phenomenon (the Weyl–Van der Pol formula). The latter arises in the problem of ...diffraction of waves generated by a source located near an impedance plane. An incident wave illuminates an impedance wedge or cone. The singular points of the wedge’s (the edge points) or cone’s (the vertex of the cone) boundary play the role of an imaginary source, giving rise to a specific boundary layer in some neighborhood of the corresponding impedance surface, provided that the surface impedance is relatively small. From the mathematical point of view, the description of the phenomenon is given by means of the far field asymptotics for the Sommerfeld integral representations of the scattered field. For small impedance of the scattering surface, the singularities describing the surface wave, which propagates from the edge (or from the vertex) along the impedance surface, may be located in a neighborhood of saddle points. The latter are responsible for a cylindrical wave from the edge of the wedge (or for a spherical wave from the vertex of the cone). As a result, the asymptotics of the Sommerfeld integral are uniformly represented by a Fresnel type integral for the wedge problem or by a parabolic cylinder type function for the cone problem.
