Internal Lifshitz tails for discrete Schrödinger operators

Najar, Hatem

doi:https://doi.org/10.1155/IJMMS/2006/91865

International Journal of Mathematics and Mathematical Sciences

On this page

Abstract References Copyright Related Articles

Open Access

Volume 2006 | Article ID 091865 | https://doi.org/10.1155/IJMMS/2006/91865

Internal Lifshitz tails for discrete Schrödinger operators

Hatem Najar¹

Received22 Mar 2005

Revised14 May 2006

Accepted06 Aug 2006

Published23 Oct 2006

Abstract

We consider random Schrödinger operators Hω acting on l2(ℤd). We adapt the technique of the periodic approximations used in (2003) for the present model to prove that the integrated density of states of Hω has a Lifshitz behavior at the edges of internal spectral gaps if and only if the integrated density of states of a well-chosen periodic operator is nondegenerate at the same edges. A possible application of the result to get Anderson localization is given.

References

M. Aizenman and S. Molchanov, “Localization at large disorder and at extreme energies: an elementary derivation,” Communications in Mathematical Physics, vol. 157, no. 2, pp. 245–278, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Probability and Its Applications, Birkhäuser, Massachusetts, 1990.
View at: Zentralblatt MATH | MathSciNet
A. Figotin, “Existence of gaps in the spectrum of periodic dielectric structures on a lattice,” Journal of Statistical Physics, vol. 73, no. 3-4, pp. 571–585, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
A. Figotin and A. Klein, “Localization phenomenon in gaps of the spectrum of random lattice operators,” Journal of Statistical Physics, vol. 75, no. 5-6, pp. 997–1021, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Fröhlich and T. Spencer, “Absence of diffusion in the Anderson tight binding model for large disorder or low energy,” Communications in Mathematical Physics, vol. 88, no. 2, pp. 151–184, 1983.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
G. M. Graf, “Anderson localization and the space-time characteristic of continuum states,” Journal of Statistical Physics, vol. 75, no. 1-2, pp. 337–346, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Kirsch, “Random Schrödinger operators. A course,” in Schrödinger Operators (Sønderborg, 1988), vol. 345 of Lecture Notes in Phys., pp. 264–370, Springer, Berlin, 1989.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
F. Klopp, “Band edge behavior of the integrated density of states of random Jacobi matrices in dimension $1$ ,” Journal of Statistical Physics, vol. 90, no. 3-4, pp. 927–947, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
F. Klopp, “Internal Lifshitz tails for random perturbations of periodic Schrödinger operators,” Duke Mathematical Journal, vol. 98, no. 2, pp. 335–396, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
I. Lifshitz, “Structure of the energy spectrum of impurity bands in disordered solid solutions,” Soviet Physics JETP, vol. 17, pp. 1159–1170, 1963.
View at: Google Scholar
G. A. Mezincescu, “Internal Lifschitz singularities for one-dimensional Schrödinger operators,” Communications in Mathematical Physics, vol. 158, no. 2, pp. 315–325, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Najar, “Lifshitz tails for random acoustic operators,” Journal of Mathematical Physics, vol. 44, no. 4, pp. 1842–1867, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Najar, “Asymptotic behavior of the integrated density of states of acoustic operators with random long range perturbations,” Journal of Statistical Physics, vol. 115, no. 3-4, pp. 977–996, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Najar, “2-dimensional localization of acoustic waves in random perturbation of periodic media,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 1–17, 2006.
View at: Publisher Site | Google Scholar
L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, vol. 297 of Fundamental Principles of Mathematical Sciences, Springer, Berlin, 1992.
View at: Zentralblatt MATH | MathSciNet
M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.
View at: Zentralblatt MATH | MathSciNet
B. Simon, “Lifschitz tails for the Anderson model,” Journal of Statistical Physics, vol. 38, no. 1-2, pp. 65–76, 1985.
View at: Publisher Site | Google Scholar | MathSciNet
B. Simon, “Internal Lifschitz tails,” Journal of Statistical Physics, vol. 46, no. 5-6, pp. 911–918, 1987.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. Stollmann, Caught by Disorder Bounded States in Random Media, Birkhäuser, Massachusetts, 2001.
I. Veselić, “Localization for random perturbation of periodic Schrödinger operators with regular Floquet eigenvalues,” Tech. Rep., University of Bochum, Bochum, 2005.
View at: Google Scholar
H. von Dreifus and A. Klein, “A new proof of localization in the Anderson tight binding model,” Communications in Mathematical Physics, vol. 124, no. 2, pp. 285–299, 1989.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

202

Downloads

646

Citations