On the Time – Dependent Approach to Anderson Localization

A simple proof of Anderson localization is obtained. This is done by giving a bound on the averaged time evolution and then ruling out the existence of any continuous spectrum for large disorder with the help of the RAGE theorem. Furthermore the decoupling lemma of Aizenman and Molchanov is extended to its natural setting, the Holder continuous measures.

[1]  F. Martinelli,et al.  Remark on the absence of absolutely continuous spectrum ford-dimensional Schrödinger operators with random potential for large disorder or low energy , 1985 .

[2]  F. Martinelli,et al.  Constructive proof of localization in the Anderson tight binding model , 1985 .

[3]  Y. Lévy,et al.  Anderson localization for multi-dimensional systems at large disorder or large energy , 1985 .

[4]  J. Fröhlich,et al.  Absence of diffusion in the Anderson tight binding model for large disorder or low energy , 1983 .

[5]  M. Aizenman,et al.  Localization at large disorder and at extreme energies: An elementary derivations , 1993 .

[6]  B. Simon,et al.  Singular continuous spectrum under rank one perturbations and localization for random hamiltonians , 1986 .

[7]  V. Enss.  Asymptotic completeness for quantum mechanical potential scattering , 1978 .

[8]  M. Aizenman.  LOCALIZATION AT WEAK DISORDER: SOME ELEMENTARY BOUNDS , 1994 .

[9]  A. Klein,et al.  A new proof of localization in the Anderson tight binding model , 1989 .

[10]  Hervé Kunz,et al.  Sur le spectre des opérateurs aux différences finies aléatoires , 1980 .

[11]  B. Simon,et al.  Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization , 1996 .

[12]  Hans L. Cycon,et al.  Schrodinger Operators: With Application to Quantum Mechanics and Global Geometry , 1987 .

[13]  René Carmona,et al.  Anderson localization for Bernoulli and other singular potentials , 1987 .

[14]  G. M. Graf.  Anderson localization and the space-time characteristic of continuum states , 1994 .