Some examples in one-dimensional ``geometric'' scattering on manifolds
type: Item
instance of: scholarly article
title: Some examples in one-dimensional ``geometric'' scattering on manifolds
zbMATH Open document ID: 0914.58037
author: Alexander Kiselev
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-06-06T00:00:00Z
review text: Let \(\Omega\) be a compact Riemannian manifold which is inserted between two half lines; this means that one glues on two copies of \([0,\infty)\) to two points \(x_1\) and \(x_2\) of a compact Riemannian manifold \(M\). The standard Laplacian \(\Delta_M\) is chosen on \(M\); the operator \(-\partial/\partial x^2\) is chosen on the two half lines. Boundary conditions are chosen at the gluing points \(x_i\) so that the Laplacian does not decompose as a direct sum with summands acting on the different geometric components. The spectrum of the resulting operator is studied as are the transition coefficient in the scattering problem. The author applies these results to the particular example in which \(M\) is the unit sphere in \(\mathbb{R}^3\) and in which the two half lines are glued on at antipodal points; spherical harmonics are used to analyze the situation and the large energy setting is studied.
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1053563
zbMATH Keywords: transition coefficient
zbMATH Keywords: scattering
zbMATH Keywords: Wannier-Stark ladders
zbMATH Keywords: sharp resonances
zbMATH Keywords: singular Laplacian
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
OpenAlex ID: W2045389405
cites work: Q3657016
cites work: Periodic Schrödinger operators with large gaps and Wannier-Stark ladders
cites work: The absence of the absolutely continuous spectrum for δ ′ Wannier–Stark ladders
cites work: Quantum motion on a half-line connected to a plane
cites work: Scattering problems on noncompact graphs
cites work: Perturbation formula of eigenvalues in singularly perturbed domain
cites work: Eigenfrequencies and eigenfunctions of the Laplacian for Neumann boundary conditions in a system of two coupled cavities
cites work: A traveling wave in a ring resonator
cites work: Acoustic model of zero-width slits and hydrodynamic boundary layer stability
cites work: THE HELMHOLTZ RESONATOR AND THE THEORY OF OPERATOR EXTENSIONS IN A SPACE WITH INDEFINITE METRIC
cites work: Q4066072
cites work: Q3900552
DOI: 10.1006/JMAA.1997.5497
description: scientific article; zbMATH DE number 1053563