On the probability of the existenceof a localized basic state for a discrete Schrödinger equation with random potential, perturbed by a compact operator
Abstract:
Let $H_d$ be the difference Laplace operator in $l_2(\mathbf{Z}^d)$ and $\mathbf{W}$ be a discrete potential (a bounded diagonal operator). We search for the conditions on the spectrum of the operator $H_d+\mathbf{W}$ under which the complete Hamiltonian $H_d+\mathbf{W}+\mathbf{V}(\omega)$ with random potential $\mathbf{V}(\omega)$ has a localized basic state (a) with positive probability and (b) with probability 1. We prove that the condition that the maximal point of the spectrum of $H_d+\mathbf{W}$ is isolated from the remaining spectral points of the operator is sufficient for (a) to be true (if $\mathbf{W}$is a compact operator this condition is necessary). Respectively, the condition that the length of the random potential does not exceed the distance between the maximal point of the spectrum of $H_d+\mathbf{W}$ and the rightmost point of its essential spectrum is a sufficient one for (b) to be true. It is shown that if $\mathbf{W}$ is an operator of rank 1, then this condition is necessary.
Keywords:indecomposable symmetric matrix, boundedself-adjoint operator, discrete Schrö, dinger equation with random potential, compact diagonal operator, operator of rank 1, translation-invariant measure, Anderson's model.
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