Helmholtz operators on harmonic manifolds
type: Item
instance of: scholarly article
title: Helmholtz operators on harmonic manifolds
zbMATH Open document ID: 0818.58042
DOI: 10.1007/BF02398435
author: Henrik Schlichtkrull
author: Rainer Schimming
published in: Acta Mathematica
publication date: 1995-08-24T00:00:00Z
review text: The authors investigate the Helmholtz operators \(L_ \lambda = \Delta + \lambda\), where \(\Delta\) is the Laplace-Beltrami operator and \(\lambda = \text{const }\in \mathbb{C}\), on a harmonic manifold \(M\), \(\dim M = n\), with the pseudo-Riemannian metric \(g\). The number \(\lambda\) is called an exceptional Helmholtz number for \((M,g)\) if \(L_ \lambda u = 0\) admits Hadamard's logarithm-free elementary solution. It is well-known that for odd \(n\) all \(\lambda\) are exceptional. The authors associate to any even- dimensional harmonic manifolds a certain polynomial of the degree \(m = {1\over 2}(n-2)\), called Hadamard polynomial. The main result of the first half of the paper is that if \(\lambda\) is exceptional then it is a root of the Hadamard polynomial and if \((M,g)\) is analytic then all roots of the Hadamard polynomial are exceptional Helmholtz numbers. In the second half the authors explicitly determine the exceptional Helmholtz numbers for the (even-dimensional) isotropic spaces.
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 719521
zbMATH Keywords: Helmholtz operator
zbMATH Keywords: Hadamard solution
zbMATH Keywords: harmonic manifolds
reviewed by: Yuri E. Gliklikh
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 719521