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

Published online by Cambridge University Press:  12 July 2007

Seok Woo Kim  and
Yong Hah Lee
Seok Woo Kim
Affiliation:
Department of Mathematics, Chosun University, Gwangju 501-759, Korea (seokwoo@mail.chosun.ac.kr)
Yong Hah Lee
Affiliation:
Department of Mathematics Education, Ewha Womans University, Seoul 120-750, Korea (yonghah@ewha.ac.kr)
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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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 133 , Issue 4 , August 2003 , pp. 855 - 873
Copyright
Copyright © Royal Society of Edinburgh 2003

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