Skip to main content
SpringerLink
Log in

A time-dependent method for computing eigenfunctions and eigenvalues of Schrödinger operators

  • Published:
Japan Journal of Applied Mathematics Aims and scope Submit manuscript

Abstract

A method is proposed to compute an eigenfunction and the associated eigenvalue of the Schrödinger operatorH=−Δ+V(x), the eigenvalue being required to be closest to a given numberE. The idea is to use a time-dependent Schrödinger equation with the inhomogeneous term exp(−irE)h to excite the target mode. A device is introduced to avoid exciting remote resonant eigenvalues. Some analysis of the schemes is given and it is shown that the use of extrapolations is effective. Some numerical examples are presented. They indicate that the proposed method produces rather accurate results.

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. M.D. Feit, J.A. Fleck, Jr., and A. Steiger, Solutions of the Schrödinger equation by a spectral method. J. Comput. Phys.,47 (1982), 412–433.

    Article  MATH  MathSciNet  Google Scholar 

  2. M.D. Feit and J.A. Fleck, Jr., Wave packet dynamics and chaos in the Hénon-Heiles system. J. Chem. Phys.,80 (1984), 2578–2584.

    Article  MathSciNet  Google Scholar 

  3. D.R. Grempel, R.E. Prange, and S. Fishman, Quantum dynamics of a nonintegrable system. Phys. Rev. A,29 (1984), 1639–1647.

    Article  Google Scholar 

  4. R. Grimm and R.G. Storer, A new method for the numerical solution of the Schrödinger equation. J. Comput. Phys.,4 (1969), 230–249.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Grimm and R.G. Storer, Improvements on a new method for the numerical solution of the Schrödinger equation. ibid,,9 (1972), 538–546.

    Article  MathSciNet  Google Scholar 

  6. D.C. Joyce, Survey of extrapolation process in numerical analysis. SIAM Rev.,13 (1971), 435–490.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Kato, Perturbation Theory for Linear Operators. 2nd ed., 1976, Springer.

  8. F. Odeh and J.B. Keller, Partial differential equations with periodic coefficients and Bloch waves in crystals. J. Math. Phys.,5 (1964), 1499–1504.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Samuelides, R. Fleckinger, L. Touzillier, and J. Bellisard, Instabilities of the quantum rotator and transition in the quasi-energy spectrum. Europhys. Lett.,1 (1986), 203–208.

    Article  Google Scholar 

  10. J. Stoer and R. Burlrisch, Introduction to Numerical Analysis. Springer, 1980.

  11. Appendix of the preprint of the present paper. To be published separately.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Gakushuin University, Mejiro, 171, Toshima-ku, Tokyo, Japan

    S. T. Kuroda

  2. Department of Mathematics, Yamanashi University, 400, Takeda, Kōfu, Japan

    Toshio Suzuki

Authors
  1. S. T. Kuroda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Toshio Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Kuroda, S.T., Suzuki, T. A time-dependent method for computing eigenfunctions and eigenvalues of Schrödinger operators. Japan J. Appl. Math. 7, 231–253 (1990). https://doi.org/10.1007/BF03167843

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167843

Key words

Search

Navigation