Periodic Schrödinger operators on a manifold. (English) Zbl 0655.58033

Kobayashi, Toshiyuki; Ono, Kaoru; Sunada, Toshikazu

Forum Math. 1, No. 1, 69-79 (1989).

The present paper treats some spectral properties of the Schrödinger operator \(-\Delta +q\) defined on an open Riemannian manifold with an isometry group \(\Gamma\) whose orbit space is compact, where the function q is supposed to be periodic under the \(\Gamma\)-action. A generalization of the Bloch’s theorem is discussed. Examples of manifolds on which the Schrödinger operators have eigenvalues are exhibited.

Reviewer: T.Kobayashi

MSC:

58J50	Spectral problems; spectral geometry; scattering theory on manifolds
35J10	Schrödinger operator, Schrödinger equation

Keywords:

spectral properties of the Schrödinger operator; Riemannian manifold; Bloch’s theorem; eigenvalues