Spectral Gaps of Schrödinger Operators on Hyperbolic Space

This paper is mainly concerned with estimates of spectral gaps of Schrödinger Operators with smooth potential on real hyperbolic space. The estimates are obtained by explicit constructions of approximate generalized eigenfunctions. Among the results are analogues of classical uniform and asymptotic gap estimates for periodic Schrödinger Operators on Euclidean space. Moreover, in the more general setting of an arbitrary complete non-compact Riemannian manifold, we derive a growth condition for a generalized eigenfunction such that the corresponding eigenvalue is also a point of the L²-spectrum.