Square integrable solutions to the Klein-Gordon equation on a manifold

The problem of finding eigenvalues and square integrable eigenfunctions was considered for the hyperbolic Klein-Gordon equation on a pseudo-Riemannian manifold. An infinite series of square integrable solutions to the Klein-Gordon equation on a Friedman-type manifold, in particular, on the de Sitter space was constructed. These solutions correspond to a discrete mass spectrum and a finite action. The condition involves integration with respect to both spatial variables and time variables, in accordance with the fact that theses variables are equivalent in a certain sense in relativity theory. The Friedman metric has a form where h ijis the Riemannian metric on a 3-manifold of constant positive, negative. or zero curvature. the function a(t) is determined from Einstein-Friedman equations. There is a well-developed spectral theory for elliptic differential operators.