The eigenvalues of the Laplacian on a sphere with boundary conditions specified on a segment of a great circle

Abstract

We prove that the eigenvalues of the Laplacian on a sphere with a Dirichlet boundary condition specified on a segment of a great circle lie between an integer and a half-integer and for a Neumann boundary condition they lie between a half integer and an integer. These eigenvalues correspond to the eigenvalues of the angular part of the Laplacian with boundary conditions specified on a plane angular sector, which are relevant in the calculation of scattering amplitude. These eigenvalues can also be used to determine the behavior of the fields near the tip of a plane angular sector as a function of the distance to the tip. The first few eigenvalues for both Dirichlet and Neumann boundary conditions are calculated. The same eigenvalues are also calculated using the Wentzel–Kramers–Brillouin (WKB) method. There is excellent agreement between the exact and the WKB eigenvalues.

Electromagnetic scattering by a plane angular sector: I. Diffraction coefficients of the spherical wave from the vertex

“…The functions g u,v (ω, ω 0 , ν) are regular near the points ν = ±1/2, because their singularities ν j (ν 2 j are the eigenvalues of the corresponding Laplace-Beltrami operators) do not coincide with ν = ±1/2 (see [35]). For the potentials of the incident wave one has…”

“…The problem for the spectral function can be efficiently studied (e.g. [35]) or can be reduced to an integral equation on a segment like in [12]. However, we give an alternative approach for reduction of the problem (37), (38) for the spectral function which is based on the separation of the angular variables and on linear summation equations for the corresponding Fourier coefficients.…”

“…In this case traditional methods of the study of the Green's function can be applied [9,11,35]. However, the Green's function  g u,v (ω, ω 0 , ν) can be also written in terms of the solutions u D ν and u N ν of the Dirichlet (for u) and Neumann (for v) problems for the equations…”

“…The poles ν j , µ j of these functions are on the real axis in the exterior of the segment (−1/2, 1/2) and, in particular, ν 2 n = c 0 n 1/2 + O(n 1/4 log n), n → ∞. It is worth to notice that algorithm of calculation of the eigenvalues and eigenfunctions is discussed in the work [35].…”

On the diffraction of acoustic waves by a quarter-plane

“…In this report, Jansen and Boersma used a different method based on Lamé functions that will not be discussed further. Other authors, such as Sleeman et al [27] and Abawi et al [28] also have computed these eigenvalues using different methods.…”