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Square integrable solutions to the Klein-Gordon equation on a manifold. (English. Russian original) Zbl 1168.58011
Dokl. Math. 73, No. 3, 441-444 (2006); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 408, No. 1, 317-320 (2006).
Let \(M\) be an \((n + 1)\) dimensional pseudo-Riemannian manifold of signature \((0, 1)\) with metric tensor \(g_{\mu, \nu}, \mu, \nu = 0, 1, \dots, n\). The authors consider the Klein-Gordon equation on \(M\) for a real function \(f\) \[ \square f + \lambda f = 0. \] Here \[ \square f = \nabla_{\mu}\nabla^{\mu}f = \frac 1{\sqrt{|g|}}\partial_{\mu}(\sqrt{|g|}g^{\mu \nu}\partial_{\nu}f, \] \(g\) is the determinant of the matrix \(\| g_{\mu \nu}\|\), and the real parameter \(\lambda\) corresponds to the square of the mass. An infinite set of square integrable solutions is constructed on Friedman-type manifolds, in particular on the de Sitter space. These solutions correspond to a discrete spectrum of masses and a finite action.

58J45 Hyperbolic equations on manifolds
35L10 Second-order hyperbolic equations
83C47 Methods of quantum field theory in general relativity and gravitational theory
35P05 General topics in linear spectral theory for PDEs
83F05 Relativistic cosmology
Full Text: DOI
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