Abstract
We prove almost-sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete Schrödinger operators on one- and quasi-one-dimensional lattices. This paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems; we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential. Our results allow us to prove that, e.g., for strong disorder, the smallest positive Lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. W. Anderson, Absence of diffusion in certain random lattices,Phys. Rev. 109:1492 (1958).
J. Avron and B. Simon, Singular continuous spectrum for a class of almost periodic Jacobi matrices,Bull. Am. Math. Soc. 6:81 (1982).
Ph. Bougerol and J. Lacroix, Product of random matrices with applications to Schrödinger operators, Birkhauser, to appear.
R. Carmona, Exponential localization in one-dimensional disordered systems,Duke Math. J. 49:191 (1982).
R. Carmona, One-dimensional Schrödinger operators with random or deterministic potentials: new spectral types,J. Funct. Anal. 51:229 (1983).
R. Carmona, Cours à St Flour, in (Lecture Notes in Mathematics, Springer, New York, to appear).
F. Delyon, Apparition of purely singular continuous spectrum in a class of random Schrödinger operators, unpublished.
F. Delyon, H. Kunz, and B. Souillard, One-dimensional wave equations in disordered media,J. Phys. A 16:25 (1983).
F. Delyon, Y.-E. Lévy, and B. Souillard, Anderson localization for multi-dimensional systems at large disorder or large energy,Commun. Math. Phys., to appear.
F. Delyon, Y.-E. Lévy, and B. Souillard, An approach “à la Borland” to multi-dimensional localization, preprint, école Polytechnique.
F. Delyon, B. Simon, and.B. Souillard, From power pure point to continuous spectrum,Ann. Inst. Henri Poincaré, to appear.
J. Fröhlich, F. Martinelli, E. Scoppola, and T. Spencer, Anderson localization for large disorder or low energy, preprint No. 426, Università di Roma.
J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy,Commun Math. Phys. 88:151 (1983).
Ya. Goldsheid, Talk at the Tachkent conference on information theory, September 1984.
Ya. Goldsheid,Dokl. Akad. Nauk. SSSR 225:273 (1980).
Ya. Goldsheid, S. Molchanov, and L. Pastur, A pure-point spectrum of the stochastic and one-dimensional Schrödinger equation,Funct. Anal. Appl. 11:1 (1977).
S. Kotani, Lyapunov exponents and spectra for one-dimensional Schrödinger operators, Proceedings of the AMS meeting on “Random Matrices,” 1984.
S. Kotani, Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators,Proc. Taniguchi Symp. SA Katata 225 (1982).
H. Kunz and R. Johnston, A method for calculating the localisation length, with an analysis of the Lloyd model,J. Phys. C 16:4565 (1983).
H. Kunz and B. Souillard, On the spectrum of random finite difference operators,Commun. Math. Phys. 78:201 (1980).
J. Lacroix, Problèmes probabilistes liés à l'étude des opérateurs aux différences aléatoires,Ann. Inst. E. Cartan, Nancy 7 (1983).
J. Lacroix, Singularités du spectre de l'opérateur de Schrödinger aléatoire dans un ruban ou un demi-ruban,Ann. Inst. Henri Poincaré 38A:385 (1983).
J. Lacroix, Localisation pour l'opérateur de Schrödinger aléatoire dans un ruban,Ann. Inst. Henri Poincaré 40A:97 (1984).
N. Minami, An extension of Kotani's theorem for random generalized Sturm-Liouville operators, preprint.
N. F. Mott and W. D. Twose, The theory of impurity conduction,Adv. Phys. 10:107 (1961).
L. Pastur, Spectral properties of disordered systems in one-body approximation,Commun. Math. Phys. 75:179 (1980).
G. Royer, étude des opérateurs de Schrödinger à potentiel aléatoire en dimension 1,Bull. Soc. Math. France,110:27 (1982).
B. Simon, Localization in general one-dimensional random systems,Commun. Math. Phys., to appear.
B. Simon, Some Jacobi matrices with decaying potential and dense point spectrum,Commun. Math. Phys. 87:253 (1983).
B. Simon, Kotani theory for one dimensional stochastic Jacobi matrices,Commun. Math. Phys. 89:227 (1983).
B. Simon and B. Souillard, Franco-American meeting on the mathematics of random and almost-periodic potential,J. Stat. Phys. 36:273 (1984).
B. Simon and T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians,Commun. Pure Appl. Math., to appear.
B. Simon, M. Taylor, and T. Wolff, Some rigorous results for the Anderson model, unpublished.
B. Souillard, Electrons in random and almost-periodic potentials, in Proceedings of “Common Trends in Particle and Condensed Matter Physics” Les Houches 1983,Phys. Rep. 103:41 (1984).
F. Wegner, Bounds on the density of states in disordered systems,Z. Phys. B 44:9 (1981).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Delyon, F., Lévy, Y. & Souillard, B. Anderson localization for one- and quasi-one-dimensional systems. J Stat Phys 41, 375–388 (1985). https://doi.org/10.1007/BF01009014
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01009014