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Communications in Mathematical Physics

, Volume 164, Issue 3, pp 489–505 | Cite as

Pure point spectrum under 1-parameter perturbations and instability of Anderson localization

  • A. Ya. Gordon
Article

Abstract

We consider a selfadjoint operator,A, and a selfadjoint rank-one projection,P, onto a vector, φ, which is cyclic forA. We study the set of all eigenvalues of the operatorA t =A+tP (t∈∝) that belong to its essential spectrum (which does not depend on the parametert). We prove that this set is empty for a dense set of values oft. Then we apply this result or its idea to questions of Anderson localization for 1-dimensional Schrödinger operators (discrete and continuous).

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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References

  1. [AG] Akhiezer, N.I., Glasman, I.M.: Theory of linear operators in Hilbert space (in 2 volumes), vol. 2. New York: Ungar Publ., 1963Google Scholar
  2. [CL] Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill 1955Google Scholar
  3. [CD] Chulayevsky, V.A., Sinai, Ya.G.: Anderson localization for the 1-D discrete Schrödinger operator with two-frequency potential. Commun. Math. Phys.125, 91–112 (1989)Google Scholar
  4. [DLS] Delyon, F., Lévy, Y., Souillard, B.: Anderson localization for one- and quasi-one-dimensional systems. J. Stat. Phys.41, No. 3/4, 375–388 (1985)Google Scholar
  5. [DS] Dunford, N., Schwarz, J.: Linear operators. Spectral theoryGoogle Scholar
  6. [Gl] Glazman, I.M.: Direct methods of qualitative spectral analysis of singular differential operators. Moscow: Nauka 1963 (in Russian)Google Scholar
  7. [Go] Gordon, A.Ya. Determinate potential with pure point spectrum. Mat. Zametki48, No. 6, 38–46 (1990) (in Russian)Google Scholar
  8. [Go'] Gordon, A.Ya. On exceptional values of the boundary phase for the Schrödinger equation on a half axis. Russian Math. Surveys.47, 211–212 (1992) (in Russian)Google Scholar
  9. [KMP] Kirsch, W., Molchanov, S.A., Pastur, L.A.: One-dimensional Schrödinger operator with an unbounded potential: Pure point spectrum. Funkt. Anal. i Prilozhen.24, No. 3 (1990) (in Russian)Google Scholar
  10. [Ko] Kotani, S.: Lyapunov exponents and spectra for one-dimensional Schrödinger operators. AMS Series of Contemp. Math.50, 277–286 (1986)Google Scholar
  11. [O] Oxtoby, J.C.: Measure and category. Berlin, Heidelberg, New York: Springer 1971Google Scholar
  12. [P] Pastur L.: Spectral properties of disordered systems in one-body approximation. Commun. Math. Phys.75, 179 (1980)Google Scholar
  13. [RJMS] Del Rio, R., Jitomirskaya, S., Makarov, N., Simon, B.: Singular continuous spectrum is generic (preprint)Google Scholar
  14. [S] Simon, B.: Spectral analysis of rank one perturbations and applications. Lecture notes. Vancouver, 1993Google Scholar
  15. [SW] Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Commun. Pure Appl. Math.39, 75–90 (1986)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • A. Ya. Gordon
    • 1
  1. 1.Warshavskoye shosseMITPANMoscowRussia

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