Journal of Operator Theory

Volume 41, Issue 1, Winter 1999 pp. 151-173.

Singular limits of Schrodinger operators and Markov processes

Authors: Y. Colin de Verdiere (1), Y. Pan (2), and B. Ycart (3)
Author institution: (1) Institut Fourier, B.P. 74, 38402 Saint Martin d'Heres Cedex, France
(2) LMC/IMAG, B.P. 53, 38041 Grenoble Cedex 9, France
(3) Bureau 39, Tour IRMA, LMC/IMAG, B.P. 53, 38041 Grenoble Cedex 9, France

Summary: After introducing the $\Gamma$-convergence of a family of symmetric matrices, we study the limits in that sense, of Schr\"odinger operators on a finite graph. The main result is that any such limit can be interpreted as a Schr\"odinger operator on a new graph, the construction of which is described explicitly. The operators to which the construction is applied are reversible, almost reducible Markov generators. An explicit method for computing an equivalent of the spectrum is described. Among possible applications, quasi-decomposable processes and low-temperature simulated annealing are studied.

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