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Singular limits of Schrödinger operators and Markov processes. (English) Zbl 0990.47013
Summary: After introducing the \(\Gamma\)-convergence of a family of symmetric matrices, we study the limits in that sense, of Schrödinger operators on a finite graph. The main result is that any such limit can be interpreted as a Schrödinger 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.

47A55 Perturbation theory of linear operators
60J27 Continuous-time Markov processes on discrete state spaces
47F05 General theory of partial differential operators
35J10 Schrödinger operator, Schrödinger equation