Abstract
We consider the lattice Schrödinger operator acting onl 2 (ℤd) with random potential (independent, identically distributed random variables), supported on a subspace of dimension 1 ⩽v <d. We use the multiscale analyses scheme to prove that this operator exhibits exponential localization at the edges of the spectrum for any disorder or outside the interval [-2d, 2d] for sufficiently high disorder.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Pastur, L. and Figotin, A.,Spectra of Random and Almost Periodic Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1992.
Carmona, R. and Lacroix, J.,Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
Fröhlich, J., Martinelli, F., Scoppola, E., and Spencer, T., Constructive proof of localization in the Anderson tight binding model,Comm. Math. Phys. 101, 21–46 (1985).
Simon, B. and Wolff, T., Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians,Comm. Pure Appl. Math.,39, 75–90 (1986).
Delyon, F., Levi, Y., and Souillard, B., Anderson localization for multidimensional systems at large disorder or low energy,Comm. Math. Phys. 100, 463–470 (1985).
Fröhlich, J. and Spencer, T., Absence of diffusion in the Anderson tight binding model for large disorder or low energy,Comm. Math. Phys. 101, 21–46 (1985).
Spencer, T., Localization for random and quasi-periodic potentials,J. Stat. Phys. 51, 1009 (1988).
von Dreifus, H., On the effects of randomness in ferromagnetic models and Schrödinger operators, NYU PhD Thesis (1987).
von Dreifus, H. and Klein, A., A new proof of localization in the Anderson tight binding model,Comm. Math. Phys. 124, 285–299 (1989).
Grinshpun, V., Point spectrum of random infinite-order operator acting onl 2 (ℤd),Rep. Ukrain. Acad. Sci. 8, 18–21 (1992) (in Russian).
Grinshpun, V., Sufficient conditions for localization of finite-difference infinite-order operator with random potential,Rep. Ukrain. Acad. Sci. 5, 26–29 (1993) (in Russian).
Grinshpun, V., Exponential localization for finite-difference infinite-order operator with random potential, to appear inMath. Phys. Anal. Geom. 2 (1994).
Grinshpun, V., Constructive proof of localization for finite-difference infinite-order operator with random potential, to appear inRandom Oper. Stochast. Equations 2 (1994).
Klopp, F., Localisation pour des opérateurs de Schrödinger aléatoires dansL 2(R d): un modèle semi-classique, Inst. Mittag-Leffler, Rep. N1, 1992-1993.
Aizenman, M. and Molchanov, S., Localization at large disorder and at extreme energies: an elementary derivation,Comm. Math. Phys. 157, 245–279 (1993).
Carmona, R., Klein, A., and Martinelli, F., Anderson localization for Bernoulli and other singular potentials.Comm. Math. Phys. 108, 41–66 (1987).