대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제25권4호
  • /
  • Pages.609-614
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  • 2010
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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ROUGH ISOMETRY AND THE SPACE OF BOUNDED ENERGY FINITE SOLUTIONS OF THE SCHRODINGER OPERATOR ON GRAPHS

  • 투고 : 2009.04.03
  • 발행 : 2010.10.31
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초록

We prove that if graphs of bounded degree are roughly isometric to each other, then the spaces of bounded energy finite solutions of the Schr$\ddot{o}$dinger operator on the graphs are isomorphic to each other. This is a direct generalization of the results of Soardi [5] and of Lee [3].

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참고문헌

  1. M. Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), no. 3, 391–413.
  2. T. Kayano and M. Yamasaki, Boundary limit of discrete Dirichlet potentials, Hiroshima Math. J. 14 (1984), no. 2, 401–406.
  3. Y. H. Lee, Rough isometry and energy finite solutions of the Schrodinger operator on graphs, Discrete Math. 263 (2003), no. 1-3, 167–177. https://doi.org/10.1016/S0012-365X(02)00576-9
  4. H. L. Royden, Harmonic functions on open Riemann surfaces, Trans. Amer. Math. Soc. 73 (1952), 40–94.
  5. P. M. Soardi, Rough isometries and Dirichlet finite harmonic functions on graphs, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1239–1248.
  6. M. Yamasaki, Ideal boundary limit of discrete Dirichlet functions, Hiroshima Math. J. 16 (1986), no. 2, 353–360.