Helmholtz operators on harmonic manifolds
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.
scientific article; zbMATH DE number 719521
scientific article