Lifshitz tails for 2-dimensional random Schrödinger operators.
type: Item
instance of: scholarly article
title: Lifshitz tails for 2-dimensional random Schrödinger operators.
zbMATH Open document ID: 1058.47034
DOI: 10.1007/BF02786575
author: Frédéric Klopp
author: Thomas H. Wolff
published in: Journal d'Analyse Mathématique
publication date: 2002-01-01T00:00:00Z
review text: The authors prove that, at an open band edge, the density of states (DOS) of 2-dimensional continuous Anderson operators exhibits Lifshitz tails. Except for the dimension 1, the previous result was obtained by \textit{F. Klopp} [Duke Math. J. 98, No. 2, 335--396 (1999); correction ibid. 109, No. 2, 411--412 (2001; Zbl 1060.82509)] where it is proved that, taking \(0\) as the bottom of the spectrum, one has the Lifshitz behaviour \(\log| \log N(E)| \sim_{E\to 0^+} -\frac d2 \log E\) if and only if the DOS \(N_{\text{per}}\) of a well chosen underlying periodic Schrödinger operator satisfies \(\log N_{\text{per}}(E)\sim_{E\to 0^+} \frac d2 \log E\) (where \(d\) is the dimension). For 2-dimensional continuous Anderson models, the article under review answers the question of the existence of Lifshitz tails when the latter auxiliary property is not satisfied. As a first result, and under fairly general hypotheses on the Anderson potential and on the random variables, the authors show that \[ \lim_{E\to E_0}\frac{\log| \log | N(E) - N(E_0| | }{\log E} < 0, \] where \(E_0\) is the edge of the band. Under some extra hypotheses, and if \(E_0\) is simple, the Lifshitz exponent (given by the limit above) is shown to be determined by the behaviour of the Floquet eigenvalues of some background periodic Schrödinger operator. In particular, the Lifshitz exponent need not be \(\frac d2 = 1\).
Mathematics Subject Classification ID: 47B80
Mathematics Subject Classification ID: 82B44
zbMATH DE Number: 1998318
zbMATH Keywords: random operator
zbMATH Keywords: Lifshitz tail
zbMATH Keywords: density of state
reviewed by: François Germinet
MaRDI profile type: Publication
cites work: Q3125297
cites work: Semianalytic and subanalytic sets
cites work: Q3997506
cites work: Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators
cites work: Analytische Faserungen über holomorph-vollständigen Räumen
cites work: Q3202580
cites work: Asymptotic of the density of states for the Schrödinger operator with periodic electric potential
cites work: Q5521308
cites work: Unique continuation and absence of positive eigenvalues for Schrödinger operators. (With an appendix by E. M. Stein)
cites work: Perturbation theory for the Schrödinger operator with a periodic potential
cites work: Several complex variables. IV. Algebraic aspects of complex analysis. Transl. from the Russian by J. Leiterer and J. Nunemacher
cites work: On the spectrum of Schrödinger operators with a random potential
cites work: Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians
cites work: Anderson localization for random Schrödinger operators with long range interactions
cites work: Localization for random perturbations of periodic Schrödinger operators
cites work: An asymptotic expansion for the density of states of a random Schrödinger operator with Bernoulli disorder
cites work: Band edge behavior of the integrated density of states of random Jacobi matrices in dimension 1
cites work: Internal Lifshits tails for random perturbations of periodic Schrödinger operators.
cites work: PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY OF STATES OF AN UNBOUNDED RANDOM JACOBI MATRIX
cites work: Lifshitz tails for random Schrödinger operators with negative singular Poisson potential
cites work: Endpoints of the spectrum of periodic operators are generically simple.
cites work: Floquet theory for partial differential equations
cites work: Sur le problème de la division
cites work: Lifschitz singularities for periodic operators plus random potentials
cites work: Internal Lifschitz singularities for one dimensional Schrödinger operators
cites work: Q4155577
cites work: Q4124053
cites work: Q4002485
cites work: Q4327561
cites work: The Newton polyhedron and oscillatory integral operators
cites work: Q4040885
cites work: Q4185363
cites work: THE SPECTRAL THEORY AND THE INDEX OF ELLIPTIC OPERATORS WITH ALMOST PERIODIC COEFFICIENTS
cites work: Q4016480
cites work: Q4722682
cites work: Q3874600
cites work: Lifshitz asymptotics via linear coupling of disorder
cites work: Q4219535
cites work: A restriction theorem for the Fourier transform
cites work: Newton polyhedra and estimation of oscillating integrals
cites work: Localization of random perturbations of periodic Schrödinger operators with regular Floquet eigenvalues
cites work: Q3123599
full work available at URL: https://doi.org/10.1007/bf02786575
OpenAlex ID: W2075209813
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description: scientific article; zbMATH DE number 1998318