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Localization for random Schrödinger operators with Poisson potential. (English) Zbl 0843.60058

Let \(\mu = (\mu_\omega)\) be a Poisson random measure with intensity \(\alpha > 0\) defined on a probability space \((\Omega, {\mathcal F}, {\mathcal P})\). Then there exists a sequence of random variables \(\{X_i(\omega)\}\) such that \(Y_{\pm 1} = X_{\pm 1}\), \(Y_n = X_n - X_{n - 1}\) and \(Y_{-n} = X_{-(n-1)} - X_{-n}\) are independent and identically distributed random variables with exponential distribution of parameter \(\alpha\) and \(\mu_\omega(B) = \#\{i : X_i(\omega) \in B\}\). Taking a single site potential \(f \in L^2(R)\) of non-negative and compact support function, define the Poisson potential \(V_\omega\) by \[ V_\omega(x) = \int f(x - y)d\mu_\omega(y) = \sum_i f(x - X_i(\omega)). \] Then \(H_\omega = -{d^2\over dx^2} + V_\omega\) is almost surely self-adjoint and metrically transitive. Hence there exist subsets \(\Sigma_{ac}\), \(\Sigma_{sc}\) and \(\Sigma_{pp}\) of \(R\) such that \(\Sigma_{ac} (H_\omega) = \Sigma_{ac}\), \(\Sigma_{sc} (H_\omega) = \Sigma_{sc}\) and \(\Sigma_{pp} (H_\omega) = \Sigma_{pp}\). The main result of this paper is to show that \(\Sigma_{ac} = \Sigma_{sc} = \emptyset\) and the eigenfunctions decay exponentially at the rate of Lyapunov exponent. The proof has been done along the Kotani’s method but, in the present case, the potential is not bounded. To overcome this, it is used the result that the operator \(H_a = -{d^2 \over dx^2} + V_a\), \(V_a = W_1(x - a) + W_2(x + a)\) satisfies the spectral averaging at positive energies, where \(W_1, W_2 \in L^1_{\text{loc}}(R)\), \(W_1 = 0\) in \((-\infty, 0)\) [resp. \(W_2 = 0\) in \((0,\infty)]\) and \(-{d^2\over dx^2} + W_1\) [resp. \(-{d^2\over dx^2} + W_2\)] is of limit point type at \(+\infty\) [resp. \(-\infty\)].

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
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References:

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