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Localization at weak disorder: some elementary bounds. (English) Zbl 0843.47039

Summary: An elementary proof is given of localization for linear operators \(H= H_0+ \lambda V\), with \(H_0\) translation invariant, or periodic, and \(V(\cdot)\) a random potential, in energy regimes which for weak disorder \((\lambda\to 0)\) are close to the unperturbed spectrum \(\sigma(H_0)\). The analysis is within the approach introduced in the recent study of localization at high disorder by the author and S. Molchanov [‘Mobility edge of random operators on the Bethe lattice’, in preparation]; the localization regimes discussed in the two works being supplementary. Included also are some general auxiliary results enhancing the method, which now yields uniform exponential decay for the matrix elements \(\langle 0|P_{[a, b]}\exp(- itH)|x)\rangle\) of the spectrally filtered unitary time evolution operators, with \([ a,b]\) in the relevant range.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47N55 Applications of operator theory in statistical physics (MSC2000)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35J10 Schrödinger operator, Schrödinger equation
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