Spectral gaps of Schrodinger operators on hyperbolic space

This paper is mainly concerned with estimates of spectral gaps of Schroding er operators Delta + q with smooth potential q on real hyperbolic space H-n . The estimates are obtained by explicit constructions of approximate gener alized eigenfunctions. Among the results are analogues (Theorems 3.1 and 4. 6) of classical uniform and asymptotic gap estimates for periodic Schroding er operators in L-2(R-n). Moreover, in the more general setting of an arbit rary complete non-compact Riemannian manifold X, we derive a growth conditi on for a generalized eigenfunction such that the corresponding eigenvalue s atisfies lambda epsilon sigma(Delta(X) + q) (Theorem 2.2).