TY - JOUR
T1 - Rough isometry and energy-finite solutions for the Schrödinger operator on Riemannian manifolds
AU - Kim, Seok Woo
AU - Lee, Yong Hah
PY - 2003
Y1 - 2003
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=0141957176&partnerID=8YFLogxK
U2 - 10.1017/s0308210500002717
DO - 10.1017/s0308210500002717
M3 - Article
AN - SCOPUS:0141957176
SN - 0308-2105
VL - 133
SP - 855
EP - 873
JO - Royal Society of Edinburgh - Proceedings A
JF - Royal Society of Edinburgh - Proceedings A
IS - 4
ER -