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Title: The spectrum minimum for random Schroedinger operators with indefinite sign potentials

Abstract

This paper sets out to study the spectral minimum for an operator belonging to the family of random Schroedinger operators of the form H{sub {lambda}}{sub ,{omega}}=-{delta}+W{sub per}+{lambda}V{sub {omega}}, where we suppose that V{sub {omega}} is of Anderson type and the single site is assumed to be with an indefinite sign. Under some assumptions we prove that there exists {lambda}{sub 0}>0 such that for any {lambda} set-membership sign [0,{lambda}{sub 0}], the minimum of the spectrum of H{sub {lambda}}{sub ,{omega}} is obtained by a given realization of the random variables.

Authors:
 [1]
  1. Departement de Mathematiques Physiques, I.P.E.I. Monastir, 5000 Monastir (Tunisia)
Publication Date:
OSTI Identifier:
20768718
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 1; Other Information: DOI: 10.1063/1.2162825; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; POTENTIALS; QUANTUM OPERATORS; RANDOMNESS; SCHROEDINGER EQUATION; SPECTRA

Citation Formats

Najar, Hatem. The spectrum minimum for random Schroedinger operators with indefinite sign potentials. United States: N. p., 2006. Web. doi:10.1063/1.2162825.
Najar, Hatem. The spectrum minimum for random Schroedinger operators with indefinite sign potentials. United States. doi:10.1063/1.2162825.
Najar, Hatem. Sun . "The spectrum minimum for random Schroedinger operators with indefinite sign potentials". United States. doi:10.1063/1.2162825.
@article{osti_20768718,
title = {The spectrum minimum for random Schroedinger operators with indefinite sign potentials},
author = {Najar, Hatem},
abstractNote = {This paper sets out to study the spectral minimum for an operator belonging to the family of random Schroedinger operators of the form H{sub {lambda}}{sub ,{omega}}=-{delta}+W{sub per}+{lambda}V{sub {omega}}, where we suppose that V{sub {omega}} is of Anderson type and the single site is assumed to be with an indefinite sign. Under some assumptions we prove that there exists {lambda}{sub 0}>0 such that for any {lambda} set-membership sign [0,{lambda}{sub 0}], the minimum of the spectrum of H{sub {lambda}}{sub ,{omega}} is obtained by a given realization of the random variables.},
doi = {10.1063/1.2162825},
journal = {Journal of Mathematical Physics},
number = 1,
volume = 47,
place = {United States},
year = {Sun Jan 15 00:00:00 EST 2006},
month = {Sun Jan 15 00:00:00 EST 2006}
}
  • Resonances of the one-dimensional Schroedinger operator are investigated, that is, the poles of the analytic extension of the corresponding scattering matrix. For a certain class of superexponentially decreasing potentials, including the Gaussian potential, the Born approximation is substantiated for the problem of localizing the poles of the scattering matrix. This makes it possible to find an asymptotic law (a quantization rule) for the distribution of these poles. For the first time, using the method developed in the paper, asymptotic formulae for resonances are obtained in the case of potentials with noncompact support. Bibliography: 15 titles.
  • The author reports on a considerable extension of the previously obtained bounds on the unintegrated density of states of random Schroedinger operators, consisting of the finite difference or the continuum Laplacian, acting on two different manifolds. The new bounds are obtained by a novel technique involving integration over the isospectral varieties of spatially cutoff versions of the operators.
  • The current-current correlation measure plays a crucial role in the theory of conductivity for disordered systems. We prove a Pastur-Shubin-type formula for the current-current correlation measure expressing it as a thermodynamic limit for random Schroedinger operators on the lattice and the continuum. We prove that the limit is independent of the self-adjoint boundary conditions and independent of a large family of expanding regions. We relate this finite-volume definition to the definition obtained by using the infinite-volume operators and the trace-per-unit volume.
  • We prove a conjecture of A. Hof, O. Knill and B. Simon [Commun. Math. Phys. {bold 174}, 149{endash}159 (1995)] by showing that the Rudin-Shapiro sequence is not {ital palindromic}, i.e., does not contain arbitrarily long palindromes. We prove actually this property for all paperfolding sequences and all Rudin-Shapiro sequences deduced from paperfolding sequences. As a consequence and as guessed by the above authors, their method cannot be used for establishing that discrete Schroedinger operators with Rudin-Shapiro potentials have a purely singular continuous spectrum. {copyright} {ital 1997 American Institute of Physics.}
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