Sur le spectre des opérateurs aux différences finies aléatoires

We study a class of random finite difference operators, a typical example of which is the finite difference Schrödinger operator with a random potential which arises in solid state physics in the tight binding approximation. We obtain with probability one, in various situations, the exact location of the spectrum, and criterions for a given part in the spectrum to be pure point or purely continuous, or for the static electric conductivity to vanish. A general formalism is developped which transforms the study of these random operators into that of the asymptotics of a multiple integral constructed from a given recipe. Finally we apply our criterions and formalism to prove that, with probability one, the one-dimensional finite difference Schrödinger operator with a random potential has pure point spectrum and developps no static conductivity.

[1]  Yoshiake Yoshioka.  On the singularity of the spectral measures of a semi-infinite random system , 1973 .

[2]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[3]  K. Ishii,et al.  Localization of Normal Modes and Energy Transport in the Disordered Harmonic Chain , 1970 .

[4]  M. Thorpe,et al.  Electronic Properties of an Amorphous Solid. I. A Simple Tight-Binding Theory , 1971 .

[5]  S A Molčanov,et al.  THE STRUCTURE OF EIGENFUNCTIONS OF ONE-DIMENSIONAL UNORDERED STRUCTURES , 1978 .

[6]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[7]  D. Thouless,et al.  Electrons in disordered systems and the theory of localization , 1974 .

[8]  Nevill Mott,et al.  The theory of impurity conduction , 1961 .

[9]  L. Pastur,et al.  A pure point spectrum of the stochastic one-dimensional schrödinger operator , 1977 .

[10]  T. P. Eggarter.  Schrödinger equation with a random potential: A functional approach , 1973 .

[11]  L. Pastur.  Spectra of Random Self Adjoint Operators , 1973 .

[12]  A. J. O'Connor.  A central limit theorem for the disordered harmonic chain , 1975 .

[13]  K. Ishii,et al.  Localization of Eigenstates and Transport Phenomena in the One-Dimensional Disordered System , 1973 .

[14]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[15]  P. Anderson.  Local moments and localized states , 1978 .

[16]  David Ruelle,et al.  A remark on bound states in potential-scattering theory , 1969 .

[17]  T. Verheggen.  Transmission coefficient and heat conduction of a harmonic chain with random masses: Asymptotic estimates on products of random matrices , 1979 .

[18]  C. Berge.  Graphes et hypergraphes , 1970 .

[19]  J. Lebowitz,et al.  Heat flow in regular and disordered harmonic chains , 1971 .

[20]  R. Borland.  The nature of the electronic states in disordered one-dimensional systems , 1963, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[21]  Sir Nevill Mott.  Electrons in glass , 1978 .

[22]  H. Furstenberg.  Noncommuting random products , 1963 .

[23]  L. Schwartz.  Radon measures on arbitrary topological spaces and cylindrical measures , 1973 .

[24]  G. Royer.  Croissance exponentielle de produits markoviens de matrices aléatoires , 1980 .

[25]  P. Anderson.  Absence of Diffusion in Certain Random Lattices , 1958 .