Localization in general one-dimensional random systems. II: Continuum Schrödinger operators
type: Item
instance of: scholarly article
title: Localization in general one-dimensional random systems. II: Continuum Schrödinger operators
zbMATH Open document ID: 0637.60080
DOI: 10.1007/BF01217682
author: B. George
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: [For Part I by the second author, see ibid. 102, 327--336 (1985; Zbl 0604.60062).] The authors establish the pure point character of the spectrum of the Schrödinger operator in \(L_ 2({\mathbb R})\) with random potential. Their proofs are based on the extension of ideas which have their roots in the work by \textit{I. Ya. Gol'dshtein}, \textit{S. Ya. Molchanov} and the reviewer [Funct. Anal. Appl. 11, 1--8 (1977; Zbl 0368.34015)], were made explicit by \textit{R. Carmona} [Duke Math. J. 49, 191--213 (1982; Zbl 0491.60058)] and especially by the first author [Stochastic analysis, Proc. Taniguchi Int. Symp., Katata \& Kyoto/Jap. 1982, North-Holland Math. Libr. 32, 225--247 (1984; Zbl 0549.60058)] and developed in powerful techniques by \textit{F. Delyon}, \textit{Y. Levy} and \textit{B. Souillard} [J. Stat. Phys. 41, 375--388 (1985)] and by the second author, \textit{M. Taylor} and \textit{T. Wolff} [Phys. Rev. Lett. 54, 1589--1592 (1985)]. According to these ideas, the main fact, additional to the positivity of the Lyapunov exponent (exponential decay of the Green function in the multidimensional case) which guarantees the pure point character of the Schrödinger operator spectrum is roughly speaking the ``sufficiently smooth'' distribution of the random potential. The efficient form of this fact was given by the first author as the absolute continuity with respect to the Lebesgue measure of the spectral measure of the operator averaged over the potential conditioned outside of a finite region. In discrete case when the potential is i.i.d. random variables it is sufficient to average over one variable, say \(q(0)\). For the continuous case of the operator \(-d^ 2/dx^ 2+q(x,\omega)\) with ergodic \(q(x,\omega)\), \(| q(x)| \leq C(\omega)(1+| x|^ 2)\), the authors prove the following Theorem 2.1: Let \((\Omega,{\mathcal F},\mu)\) be the probability space carrying the potential and \(\phi(x,\lambda,\theta)\) the phase of the Cauchy solution \(y(x,\lambda,\theta)\) of the…
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 34B20
zbMATH DE Number: 4038956
zbMATH Keywords: Schrödinger equation with random potential
zbMATH Keywords: pure point spectrum
zbMATH Keywords: localization
zbMATH Keywords: exponential decay of the Green function
zbMATH Keywords: Lyapunov exponent
MaRDI profile type: Publication
cites work: Exponential localization in one dimensional disordered systems
cites work: One-dimensional Schrödinger operators with random or deterministic potentials: New spectral types
cites work: Q5848416
cites work: Anderson localization for multi-dimensional systems at large disorder or large energy
cites work: Localization for off-diagonal disorder and for continuous Schrödinger operators
cites work: Constructive proof of localization in the Anderson tight binding model
cites work: A pure point spectrum of the stochastic one-dimensional Schrödinger operator
cites work: On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on \(L^ 2({\mathbb{R}}^ v)^+\)
cites work: Q3703043
cites work: Q3792002
cites work: Sur le spectre des opérateurs aux différences finies aléatoires
cites work: Remark on the absence of absolutely continuous spectrum for d-dimensional Schrödinger operators with random potential for large disorder or low energy
cites work: Q5516130
cites work: Q4066072
cites work: Schrödinger semigroups
cites work: Localization in general one dimensional random systems. I: Jacobi matrices
cites work: Singular continuous spectrum under rank one perturbations and localization for random hamiltonians
cites work: Support theorems for random Schrödinger operators
full work available at URL: https://doi.org/10.1007/bf01217682
OpenAlex ID: W2914320231
Recommended article: Schrödinger operators with an electric field and random or deterministic potentials
Recommended article: Q4356273
Recommended article: Q3484125
Recommended article: Exponential and super-exponential localizations for one-dimensional Schrödinger operators with Lévy noise potentials
Recommended article: Regularity of the invariant measure and of the density of states in the one-dimensional Anderson model
Recommended article: One-dimensional Schrödinger operators with random decaying potentials
Recommended article: On localization in the continuous Anderson-Bernoulli model in higher dimension
Recommended article: Localization for some continuous, random Hamiltonians in \(d\)-dimensions
Recommended article: LOCALIZATION FOR ONE DIMENSIONAL LONG RANGE RANDOM HAMILTONIANS
description: scientific article; zbMATH DE number 4038956