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

Spectral theory for hyperbolic equations is considered. Applications in mathematical and theoretical physics are discussed.
The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. We shall consider such a problem for the hyperbolic Klein–Gordon equation on Lorentzian manifolds. An infinite family of square integrable solutions for the Klein–Gordon equation on the Friedman type manifolds is constructed. These solutions have a discrete mass spectrum and a finite action. In particular the solutions on de Sitter space are investigated. The investigation have applications in theory of elementary particles and cosmology.