Skip to main content
SpringerLink
Log in

Krylov subspace spectral methods for the time-dependent Schrödinger equation with non-smooth potentials

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

This paper presents modifications of Krylov Subspace Spectral (KSS) Methods, which build on the work of Gene Golub and others pertaining to moments and Gaussian quadrature to produce high-order accurate approximate solutions to the time-dependent Schrödinger equation in the case where either the potential energy or the initial data is not a smooth function. These modifications consist of using various symmetric perturbations to compute off-diagonal elements of functions of matrices. It is demonstrated through analytical and numerical results that KSS methods, with these modifications, achieve the same high-order accuracy and possess the same stability properties as they do when applied to parabolic problems, even though the solutions to the Schrödinger equation do not possess the same smoothness.

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. Abdallah, N.B., Pinaud, O.: A mathematical model for the transient evolution of a resonant tunneling diode. C. R. Math. Acad. Sci. Paris 334, 283–288 (2002)

    MATH  MathSciNet  Google Scholar 

  2. Aho, A.V., Sethi, R., Ullman, J.D.: Compilers: Principles, Techniques and Tools. Addison-Wesley, Reading (1988)

    Google Scholar 

  3. Antoine, X., Besse, C.: Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation. J. Comput. Phys. 188, 157–175 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Arnold, A.: Mathematical concepts of open quantum boundary conditions. Transp. Theory Stat. Phys. 30(4–6), 561–584 (2001)

    Article  MATH  Google Scholar 

  5. Arnold, A., Ehrhardt, M., Sofronov, I.: Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability. Comm. Math. Sci. 1(3), 501–556 (2003)

    MATH  MathSciNet  Google Scholar 

  6. Atkinson, K.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1989)

    MATH  Google Scholar 

  7. Baskakov, V.A., Popov, A.V.: Implementation of transparent boundaries for numerical solution of the Schrödinger equation. Wave Motion 14, 123–128 (1991)

    Article  MathSciNet  Google Scholar 

  8. Berenger, J.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bruneau, C.H., Menza, L.D.: Conditions aux limites transparentes et artificielles pour lequation de Schrödinger en dimension 1 despace. C. R. Acad. Sci. Paris, Ser. I 320, 89–94 (1995)

    MATH  Google Scholar 

  10. Candes, E., Demanet, L., Ying, L.: Fast computation of Fourier integral operators. SIAM J. Sci. Comput. 29(6), 2464–2493 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dahlquist, G., Eisenstat, S.C., Golub, G.H.: Bounds for the error of linear systems of equations using the theory of moments. J. Math. Anal. Appl. 37, 151–166 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  12. Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  13. Gerschgorin, S.: Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk 7, 749–754 (1931)

    Google Scholar 

  14. Golub, G.H.: Some modified matrix eigenvalue problems. SIAM Rev. 15, 318–334 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  15. Golub, G.H.: Bounds for matrix moments. Rocky Mt. J. Math. 4, 207–211 (1974)

    Article  MathSciNet  Google Scholar 

  16. Golub, G.H., Meurant, G.: Matrices, moments and quadrature. In: Griffiths, D.F., Watson, G.A. (eds.) Proceedings of the 15th Dundee Conference, June–July 1993. Longman Scientific & Technical, Harlow (1994)

    Google Scholar 

  17. Golub, G.H., Welsch, J.: Calculation of Gauss quadrature rules. Math. Comput. 23, 221–230 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  18. Guidotti, P., Lambers, J.V., Sølna, K.: Analysis of 1-D wave propagation in inhomogeneous media. Numer. Funct. Anal. Optim. 27, 25–55 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hellums, J.R., Frensley, W.R.: Non-Markovian open-system boundary conditions for the time-dependent Schrödinger equation. Phys. Rev. B 49, 2904–2906 (1994)

    Article  Google Scholar 

  20. Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  21. Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1996)

    Article  MathSciNet  Google Scholar 

  22. Lambers, J.V.: Derivation of high-order spectral methods for time-dependent PDE using modified moments. Electron. Trans. Numer. Anal. 28, 114–135 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Lambers, J.V.: Implicitly defined high-order operator splittings for parabolic and hyperbolic variable-coefficient PDE using modified moments. Int. J. Comput. Sci. 2(3), 376–401 (2008)

    Google Scholar 

  24. Lambers, J.V.: Krylov subspace methods for variable-coefficient initial-boundary value problems. Ph.D. thesis, Stanford University, SCCM Program (2003). Available at http://sccm.stanford.edu/pub/sccm/theses/James_Lambers.pdf

  25. Lambers, J.V.: Krylov subspace spectral methods for variable-coefficient initial-boundary value problems. Electron. Trans. Numer. Anal. 20, 212–234 (2005)

    MATH  MathSciNet  Google Scholar 

  26. Lambers, J.V.: Practical implementation of Krylov subspace spectral methods. J. Sci. Comput. 32, 449–476 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mallo, I.A., Reguera, N.: Weak ill-posedness of spatial discretizations of absorbing boundary conditions for Schrödinger-type equations. SIAM J. Numer. Anal. 40, 134–158 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Mayfield, B.: Non-local boundary conditions for the Schrödinger equation. Ph.D. thesis, University of Rhode Island, Providence (1989)

  29. Moore, W.J.: Schrödinger: Life and Thought. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  30. Oskooi, A.F., Zhang, L., Avniel, Y., Johnson, S.G.: The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers. Opt. Expr. 16, 11376–11392 (2008)

    Article  Google Scholar 

  31. Papadakis, J.S.: Impedance formulation of the bottom boundary condition for the parabolic equation model in underwater acoustics. NORDA Parabolic Equation Workshop, NORDA Tech. Note 143 (1982)

  32. Schädle, A.: Non-reflecting boundary conditions for the two dimensional Schrödinger equation. Wave Motion 35, 181–188 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Schmidt, F., Deuflhard, P.: Discrete transparent boundary conditions for the numerical solution of Fresnel’s equation. Comput. Math. Appl. 29, 53–76 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  34. Shampine, L.F., Reichelt, M.W.: The Matlab ODE suite. SIAM J. Sci. Comput. 18, 1–22 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Energy Resources Engineering, Stanford University, Stanford, CA, 94305-2220, USA

    James V. Lambers

Authors
  1. James V. Lambers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James V. Lambers.

Additional information

Dedicated to the memory of Gene H. Golub, 1932–2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lambers, J.V. Krylov subspace spectral methods for the time-dependent Schrödinger equation with non-smooth potentials. Numer Algor 51, 239–280 (2009). https://doi.org/10.1007/s11075-009-9278-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-009-9278-z

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation