Communications in Mathematical Physics

, Volume 107, Issue 1, pp 21–28 | Cite as

Asymptotic completeness for a new class of Stark effect Hamiltonians

  • Arne Jensen
Article
Received:
Revised:

Abstract

Existence and completeness of the wave operators is shown for the Stark effect Hamiltonian in one dimension with a potentialV =W″, whereW is a bounded function with four bounded derivatives. This class of potentials include some almost periodic functions and periodic functions with average zero over a period (Stark-Wannier Hamiltonians). In the last section we discuss classical particle scattering for the same class of potentials.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Periodic Function 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agler, J., Froese, R.: Existence of Stark ladder resonances. Commun. Math. Phys.100, 161–171 (1985)Google Scholar
  2. 2.
    Avron, J. E., Herbst, I. W.: Spectral and scattering theory for Schrödinger operators related to the Stark effect. Commun. Math. Phys.52, 239–254 (1977)Google Scholar
  3. 3.
    Ben-Artzi, M.: Remarks on Schrödinger operators with an electric field and deterministic potentials. J. Math. Anal. Appl.109, 333–339 (1985)Google Scholar
  4. 4.
    Bentosela, F., Carmona, R., Duclos, P., Simon, B., Souillard, B., Weder, R.: Schrödinger operators with an electric field and random or deterministic potentials. Commun. Math. Phys.88, 387–397 (1983)Google Scholar
  5. 5.
    Dunford, N., Schwartz, J. T.: Linear operators II. New York: Wiley 1963Google Scholar
  6. 6.
    Herbst, I. W.: Unitary equivalence of Stark effect Hamiltonians. Math. Z.155, 55–70 (1977)Google Scholar
  7. 7.
    Jensen, A., Mourre, E., Perry, P.: Multiple commutator estimates and resolvent smoothness in quantum scattering theory. Ann. Inst. H. Poincaré, Sect. A,41, 207–225 (1984)Google Scholar
  8. 8.
    Mourre, E.: Link between the geometrical and the spectral transformation approaches in scattering theory. Commun. Math. Phys.68, 91–94 (1979)Google Scholar
  9. 9.
    Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391–408 (1981)Google Scholar
  10. 10.
    Perry, P. A.: Scattering theory by the Enss method. Math. Rep.1, part 1, 1983Google Scholar
  11. 11.
    Rejto, P. A., Sinha, K.: Absolute continuity for a 1-dimensional model of the Stark-Hamiltonian. Helv. Phys. Acta49, 389–413 (1976)Google Scholar
  12. 12.
    Simon, B.: Phase space analysis of simple scattering systems: Extensions of some work of Enss. Duke Math. J.46, 119–168 (1979)Google Scholar
  13. 13.
    Simon, B.: Trace ideals and their applications. Cambridge: Cambridge University Press 1977Google Scholar
  14. 14.
    Yajima, K.: Spectral and scattering theory for Schrödinger operators with Stark-effect. J. Fac. Sci. Univ. Tokyo, Sec IA,26, 377–390 (1979)Google Scholar
  15. 15.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III Scattering Theory. New York: Academic Press 1979Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Arne Jensen
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA
  2. 2.Matematisk InstitutAarhus UniversitetAarhus CDenmark

Personalised recommendations