Bounded solutions of the Schrödinger equation on noncompact Riemannian manifolds
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- Grigor'yan, A.A. J Math Sci (1990) 51: 2340. doi:10.1007/BF01094993
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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.