ROUGH ISOMETRY AND THE SPACE OF BOUNDED ENERGY FINITE SOLUTIONS OF THE SCHRODINGER OPERATOR ON GRAPHS Kim, Seok-Woo; Lee, Yong-Hah; Yoon, Joung-Hahn;
We prove that if graphs of bounded degree are roughly isometric to each other, then the spaces of bounded energy finite solutions of the Schrdinger operator on the graphs are isomorphic to each other. This is a direct generalization of the results of Soardi  and of Lee .
infinite graph; operator;bounded energy finite solution;rough isometry;
M. Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), no. 3, 391–413.
T. Kayano and M. Yamasaki, Boundary limit of discrete Dirichlet potentials, Hiroshima Math. J. 14 (1984), no. 2, 401–406.
Y. H. Lee, Rough isometry and energy finite solutions of the Schrodinger operator on graphs, Discrete Math. 263 (2003), no. 1-3, 167–177.
H. L. Royden, Harmonic functions on open Riemann surfaces, Trans. Amer. Math. Soc. 73 (1952), 40–94.
P. M. Soardi, Rough isometries and Dirichlet finite harmonic functions on graphs, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1239–1248.
M. Yamasaki, Ideal boundary limit of discrete Dirichlet functions, Hiroshima Math. J. 16 (1986), no. 2, 353–360.