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Generalized Liouville property for Schrödinger operator on Riemannian manifolds

Title
Generalized Liouville property for Schrödinger operator on Riemannian manifolds
Authors
Kim S.W.Lee Y.H.
Ewha Authors
이용하
SCOPUS Author ID
이용하scopus
Issue Date
2001
Journal Title
Mathematische Zeitschrift
ISSN
0025-5874JCR Link
Citation
vol. 238, no. 2, pp. 355 - 387
Indexed
SCI; SCIE; SCOPUS WOS scopus
Abstract
In this paper, we prove that the dimension of the space of positive (bounded, respectively) ℒ-harmonic functions on a complete Riemannian manifold with ℒ-regular ends is equal to the number of ends (ℒ-nonparabolic ends, respectively). This result is a solution of an open problem of Grigor'yan related to the Liouville property for the Schrödinger operator ℒ. We also prove that if a given 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, respectively,) 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, respectively). This is a partial answer of the question, suggested by Li, related to the regularity of ends of a complete Riemannian manifold. Especially, our results directly generalize various earlier results of Yau, of Li and Tam, of Grigor'yan, and of present authors, but with different techniques that the peculiarity of the Schrödinger operator demands.
DOI
10.1007/s002090100257
Appears in Collections:
사범대학 > 수학교육과 > Journal papers
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