dc.description.abstract | The Laplace-Beltrami operator on a sphere with a cut arises when considering
the problem of wave scattering by a quarter-plane. Recent methods developed
for sound-soft (Dirichlet) and sound-hard (Neumann) quarter-planes rely on an a
priori knowledge of the spectrum of the Laplace-Beltrami operator. In this paper
we consider this spectral problem for more general boundary conditions, including
Dirichlet, Neumann, real and complex impedance, where the value of the impedance
varies like $\textit{α/=r, r}$ being the distance from the vertex of the quarter-plane and α being
constant, and any combination of these. We analyse the corresponding eigenvalues
of the Laplace-Beltrami operator, both theoretically and numerically. We show
in particular that when the operator stops being self-adjoint, its eigenvalues are
complex and are contained within a sector of the complex plane, for which we provide
analytical bounds. Moreover, for impedance of small enough modulus |α|, the complex
eigenvalues approach the real eigenvalues of the Neumann case. | |