Spectral Localization by Gaussian Random Potentials in Multi-Dimensional Continuous Space

Details Spectral Localization by Gaussian Random Potentials in Multi-Dimensional Continuous Space Spectral Localization by Gaussian Random Potentials in Multi-Dimensional Continuous Space

1999-12-31

arxiv.org/abs/math-ph/9912025v3

J. Stat. Phys., vol. 101, pp. 935-985 (2000)

Physics

Mathematical Physics

LaTeX, 54 pages, 4 figures; final version, to appear in slightly different form in Journal of Statistical Physics

Scientific paper

10.1023/A:1026425621261

A detailed mathematical proof is given that the energy spectrum of a non-relativistic quantum particle in multi-dimensional Euclidean space under the influence of suitable random potentials has almost surely a pure-point component. The result applies in particular to a certain class of zero-mean Gaussian random potentials, which are homogeneous with respect to Euclidean translations. More precisely, for these Gaussian random potentials the spectrum is almost surely only pure point at sufficiently negative energies or, at negative energies, for sufficiently weak disorder. The proof is based on a fixed-energy multi-scale analysis which allows for different random potentials on different length scales.

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