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Random Hamiltonians ergodic in all but one direction. (English) Zbl 0691.60053

Summary: Let \(V_{\omega}^{(1)}\) and \(V_{\omega}^{(2)}\) be two ergodic random potentials on \({\mathbb{R}}^ d\). We consider the Schrödinger operator \(H_{\omega}=H_ 0+V_{\omega}\), with \(H_ 0=-\Delta\) and for \(x=(x_ 1,...,x_ d)\) \[ V_{\omega}(x)=V_{\omega}^{(1)}(x)\quad if\quad x_ 1<0,\quad and\quad =V_{\omega}^{(2)}(x)\quad if\quad x_ 1\geq 0. \] We prove certain ergodic properties of the spectrum for this model, and express the integrated density of states in terms of the density of states of the stationary potentials \(V_{\omega}^{(1)}\) and \(V_{\omega}^{(2)}\). Finally we prove the existence of the density of surface states for \(H_{\omega}\).

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
81P20 Stochastic mechanics (including stochastic electrodynamics)
Full Text: DOI

References:

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