, Volume 51, Issue 3, pp 2340-2349

Bounded solutions of the Schrödinger equation on noncompact Riemannian manifolds

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Abstract

Necessary and sufficient geometric conditions are proved for the equation Δu−Q(x)u=0, Q(x)≥0, to have a bounded nontrivial solution on a noncompact Riemannian manifold. The results imply as corollaries conditions for parabolicity and stochastic completeness of a manifold, previously established by other methods.

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 66–77, 1989.