Communications in Mathematical Physics

, Volume 88, Issue 3, pp 387–397 | Cite as

Schrödinger operators with an electric field and random or deterministic potentials

  • F. Bentosela
  • R. Carmona
  • P. Duclos
  • B. Simon
  • B. Souillard
  • R. Weder


We prove that the Schrödinger operatorH=−d2/dx2+V(x)+F·x has purely absolutely continuous spectrum for arbitrary constant external fieldF, for a large class of potentials; this result applies to many periodic, almost periodic and random potentials and in particular to random wells of independent depth for which we prove that whenF=0, the spectrum is almost surely pure point with exponentially decaying eigenfunctions.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Large Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramovitz, M., Stegun, I. A.: Handbook of mathematical functions. Dover: N.B.S. 1965Google Scholar
  2. 2.
    Avron, Y., Herbst, I.: Spectral and scattering theory of Schrödinger operators related to the Stark effect. Commun. Math. Phys.52, 239–254 (1977)Google Scholar
  3. 3.
    Carmona, R.: Exponential localization in one dimensional disorders systems. Duke Math. J.49, 191–213 (1982)Google Scholar
  4. 4.
    Craig, W., Simon, B.: Subharmonicity of the Liapunov index (to be published)Google Scholar
  5. 5.
    Dunford, N., Schwarz, J. T.: Linear operators. II. New York: Wiley 1963Google Scholar
  6. 6.
    Goldsheid, I. Ja., Molčanov, S. A., Pastur, L. A.: A pure point spectrum of the stochastic one dimensional Schrödinger equation. Funct. Anal. Appl.11, 1–10 (1977)Google Scholar
  7. 7.
    Herbst, I., Howland, J.: The Stark ladder and other one-dimensional external field problems. Commun. Math. Phys.80, 23 (1981)Google Scholar
  8. 8.
    Hörmander, L.: Hypoelliptic differential equations of second order. Acta Mathematica119, 147–171 (1967)Google Scholar
  9. 9.
    Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys.78, 201–246 (1980)Google Scholar
  10. 10.
    Molčanov, S. A.: The structure of eigenfunctions of one dimensional unordered structures. Math. USSR Izv.12, 69–101 (1978)Google Scholar
  11. 11.
    Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391–408 (1981)Google Scholar
  12. 12.
    Stone, M. H.: Linear transformations in Hilbert space and their applications to analysis. Providence: Am. Math. Soc. Coll. Publ.15, 1932Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • F. Bentosela
    • 1
  • R. Carmona
    • 2
  • P. Duclos
    • 3
    • 1
  • B. Simon
    • 4
  • B. Souillard
    • 5
  • R. Weder
    • 6
  1. 1.Département de Physique de l'Université de Luminy Marseille, and Centre de Physique Théorique, CNRSMarseilleFrance
  2. 2.Department of MathematicsUniversity of California at IrvineIrvineUSA
  3. 3.Département de MathématiquesUniversité de Toulon et du VarLa Garde
  4. 4.Department of Mathematics and PhysicsCalifornia Institute of TechnologyPassadenaUSA
  5. 5.Centre de Physique Théorique, Ecole PolytechniquePalaiseauFrance
  6. 6.IIMAS, Universidad Nacional Autonoma de MexicoMexico 20 D.F.Mexico

Personalised recommendations