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Spectral and dynamical properties of random models with nonlocal and singular interactions
Author(s) -
Hislop Peter D.,
Kirsch Werner,
Krishna M.
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310261
Subject(s) - mathematics , moment generating function , moment (physics) , function (biology) , type (biology) , wavelet , hilbert space , pure mathematics , spectral properties , statistical physics , random variable , quantum mechanics , statistics , physics , computer science , biology , astrophysics , ecology , evolutionary biology , artificial intelligence
We give a spectral and dynamical description of certain models of random Schrödinger operators on L 2 (ℝ d ) for which a modified version of the fractional moment method of Aizenman and Molchanov [3] can be applied. One family of models includes Schrödinger operators with random nonlocal interactions constructed from multidimensional wavelet bases. The second family includes Schrödinger operators with random singular interactions randomly located on sublattices of ℤ d , for d = 1, 2, 3. We prove that these models are amenable to Aizenman‐Molchanov‐type analysis of the Green's function, thereby eliminating the use of multiscale analysis. The basic technical result is an estimate on the expectation of fractional moments of the Green's function. Among our results, we prove a Wegner estimate, Hölder continuity of the integrated density of states, and spectral and Hilbert‐Schmidt dynamical localization at negative energies. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)