Skip to main content
SpringerLink
Log in

On Levels of a Weakly Perturbed Periodic Schrödinger Operator

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the Schrödinger operator with a periodic potential perturbed by a function which is periodic in two variables and exponentially decreases in third variable. When the perturbation is small it is proved that the levels (eigenvalues or resonances) exist near the stationary points of the eigenvalues of the periodic Schrödinger operator in the cell with respect to the third component of quasimomentum. The behaviour of these levels is investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. New York: Academic Press, 1978

  2. Davies, E.B.: Scattering from infinite sheet. Proc. Cambr. Philos.Soc. 82, 327–334 (1977)

    MATH  Google Scholar 

  3. Davies, E.B., Simon B.: Scattering theory for systems with different spatial asymptotics on the left and right. Commun. Math. Phys. 63, 277–301 (1978)

    MathSciNet  MATH  Google Scholar 

  4. Chuburin, Yu.P.: Solutions of the Schrödinger equation in the case of a semiinfinite crystal. Theor. Math. Phys. 98, 27–33 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Chuburin, Yu.P.: On small perturbations of the Schrödinger operator with a periodic potential. Theor. Math. Phys. 110, 351–359 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Albeverio, S., Gesztesy, F., H∅egh Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. New York - Berlin - Heidelberg: Springer-Verlag, 1988

  7. Simon, B.: The bound state of weakly coupled Schrödinger operators in one and two dimensions. Ann. Phys. 97, 279–288 (1976)

    MATH  Google Scholar 

  8. Chuburin, Yu.P.: On the Schrödinger operator with a small potential in the case of a crystal film. Math.Notes 52, 852–856 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Thomas, L.E.: Time dependent approach to scattering from impurities in a crystal. Commun. Math. Phys. 33, 335–343 (1973)

    Google Scholar 

  10. Gunning, R., Rossi, H.: Analytic Functions of Several Complex Variables. New York: Prentice-Hall, 1965

  11. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. New York: Academic Press, 1975

  12. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. Scattering Theory. New York: Academic Press, 1979

  13. Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. Berlin-Heidelberg-New York: Springer-Verlag, 1987

  14. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis. New York: Academic Press, 1972

  15. Chuburin, Yu.P.: Schrödinger operator eigenvalue (resonance) on a zone boundary. Theor. Math. Phys. 126, 161–168 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Udmurt State University, Universitetskaya st., 1, 426034, Izhevsk, Russia

    Yu.P. Chuburin

Authors
  1. Yu.P. Chuburin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu.P. Chuburin.

Additional information

Communicated by B. Simon

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chuburin, Y. On Levels of a Weakly Perturbed Periodic Schrödinger Operator. Commun. Math. Phys. 249, 497–510 (2004). https://doi.org/10.1007/s00220-004-1117-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1117-4

Keywords

Search

Navigation