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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.

##### MSC:
 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
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##### References:
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