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A new proof of localization in the Anderson tight binding model. (English) Zbl 0698.60051

The \(d\)-dimensional random Schrödinger operator \(H\) acts on a sequence \(\{\psi (n)\}\in \ell^ 2({\mathbb{Z}}^ d)\) by the formula \[ [H(\omega)\psi](n)=-\sum_{| m-n| =1}\psi (m)+\beta V(n,\omega)\psi (n), \] where \(\beta\) is some non zero real number. One assumes that \(\{V(n),\;n\in {\mathbb Z}^ d\}\) is an i.i.d. sequence of real random variables defined on a probability space (\(\Omega\), \({\mathcal F}, {\mathbb P})\). When \(d=1\) and the distribution of potentials is not concentrated on a single point and has a finite positive moment it is known that for \({\mathbb P}\) almost any \(\omega\) the spectrum of \(H(\omega)\) is pure point with exponentially decaying eigenfunctions.
The situation seems much more complicated for \(d>1\) and very few is known. The first multi-dimensional result has been established by J. Fröhlich and T. Spencer [Commun. Math. Phys. 88, 151–184 (1983; Zbl 0519.60066)] who proved the almost sure exponential decay of the Green function at a fixed energy, assuming either the distribution of potentials Hölder continuous and the disorder large enough (i.e. \(| \beta |\) large enough) or a large energy. In this situation it follows by routine arguments that if we assume that the distribution of the potentials is absolutely continuous then the spectrum is \({\mathbb{P}}\) almost surely pure point with exponentially decaying eigenfunctions.
Using methods developed by the first author [e.g. in his Ph. D. thesis] and by J. Fröhlich, F. Martinelli, E. Scoppola and T. Spencer [Commun. Math. Phys. 101, 21–46 (1985; Zbl 0573.60096)] (known as the multiscale analysis) the authors prove the same result without any smoothness assumption on the distribution of the potentials but they have to replace it by what is called a “Wegner estimate”. The key idea is to prove the exponential decay of the Green function in boxes for a whole interval of energy and this allows to skip the standard part involving the absolute continuity of the distribution of the potentials.
Reviewer: J.Lacroix

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47A10 Spectrum, resolvent
47B80 Random linear operators
47N55 Applications of operator theory in statistical physics (MSC2000)
60H25 Random operators and equations (aspects of stochastic analysis)
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