Rough isometry and energy-finite solutions for the Schrödinger operator on Riemannian manifolds

Seok Woo Kim, Yong Hah Lee

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

In this paper, we prove that the dimension of the space of bounded energy-finite solutions for the Schrödinger operator is invariant under rough isometries between complete Riemannian manifolds satisfying the local volume condition, the local Poincaré inequality and the local Sobolev inequality. We also prove that the dimension of the space of bounded harmonic functions with finite Dirichlet integral is invariant under rough isometries between complete Riemannian manifolds satisfying the same local conditions. These results generalize those of Kanai, Grigor'yan, the second author, and Li and Tam.

Original languageEnglish
Pages (from-to)855-873
Number of pages19
JournalRoyal Society of Edinburgh - Proceedings A
Volume133
Issue number4
DOIs
StatePublished - 2003

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