Generalized Liouville property for Schrödinger operator on Riemannian manifolds (Q5950067)

By generalizing some results of Yau, Li, and Tam, the present paper shows that the dimension of the space of positive (bounded, resp.) \(L\)-harmonic functions on a complete Riemannian manifold with \(L\)-regular ends is equal to the number of ends \((L\)-nonparabolic ends, resp.), and also that if a complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincaré inequality and the finite covering condition on each end, then the dimension of the space of positive (bounded, resp.) solutions for the Schrödinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends \((L\)-nonparabolic ends, resp.).