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A sinc-method computation for eigenvalues of Schrödinger operators with eigenparameter-dependent boundary conditions

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Abstract

We implement the sinc method to compute the eigenvalues of a second order boundary value problem with mixed type boundary conditions where the eigenparameter appears linearly in the boundary conditions. We investigate the behavior of the solutions as well as the characteristic determinant via successive iterations. The method is implemented by splitting the characteristic determinant into two parts, where it is proved that the unknown one lies in a Paley-Wiener space and it is approximated by an interpolation sampling theorem. Examples are illustrated numerically and graphically.

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References

  1. Annaby, M.H., Asharabi, R.M.: On sinc-based method in computing eigenvalues of boundary-value problems. SIAM J. Numer. Anal. 46, 671–690 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Annaby, M.H., Asharabi, R.M.: Truncation, amplitude, and jitter errors on \(\mathbb{R}\) for sampling series derivatives. J. Approx. Theory 163, 336–362 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Annaby, M.H., Tharwat, M.M.: On computing eigenvalues of second-order linear pencils. IMA J. Numer. Anal. 27, 366–380 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Annaby, M.H., Tharwat, M.M.: Sinc-based computations of eigenvalues of Dirac systems. BIT 47, 699–713 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boumenir, A.: The sampling method for Sturm-Liouville problems with the eigenvalue parameter in the boundary condition. Numer. Funct. Anal. Optim. 21, 67–75 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Butzer, P.L., Schmeisser, G., Stens, R.L.: An introduction to sampling analysis. In: Marvasti, F. (ed.) Non uniform sampling: theory and practices, pp. 17–121. Kluwer, New York (2001)

    Chapter  Google Scholar 

  7. Butzer, P.L., Splettstösser, W., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahresber. Deutsch. Math.-Verein. 90, 1–70 (1988)

    MathSciNet  MATH  Google Scholar 

  8. Chadan, K., Sabatier, P.C.: Inverse problems in quantum scattering theory, 2nd edn. Springer, New York (1989)

    Book  MATH  Google Scholar 

  9. Chanane, B.: Computing the spectrum of non self-adjoint Sturm-Liouville problems with parameter dependent boundary conditions. J. Comput. Appl. Math. 206, 229–237 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chanane, B.: Computation of eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using reularized sampling method. Math. Comput. 74, 1793–1801 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eastham, M.S.P.: Theory of ordinary differential equations. Van Nostrand Reinhold, London (1970)

    MATH  Google Scholar 

  12. Jagerman, D.: Bounds for truncation error of the sampling expansion. SIAM. J. Appl. Math. 14, 714–723 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kemp, R.R.D.: Operators on \(L^{2}\oplus \mathbb{C}^{r}\). Can. J. Math. 39, 33–53 (1987)

    Article  Google Scholar 

  14. Kemp, R.R.D., Lee, S.J.: Finite dimensional perturbations of differential expressions. Can. J. Math. 28, 1082–1104 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lund, J., Bowers, K.: Sinc methods for quadrature and differential equations. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  16. Naimark, M.A.: Linear differential operators. George Harrap, London (1967)

    MATH  Google Scholar 

  17. Pryce, J.D.: A Test Package for Sturm-Liouville Solvers. ACM Trans. Math. Softw. 25, 21–57 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Stenger, F.: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev. 23, 165–224 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Stenger, F.: Numerical methods based on sinc and analytic functions. Springer, New York (1993)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Cairo University, P.O. Box 12613, Giza, Egypt

    M. H. Annaby

  2. Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia

    M. M. Tharwat

Authors
  1. M. H. Annaby
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  2. M. M. Tharwat
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Correspondence to M. M. Tharwat.

Additional information

M. M. Tharwat: On leave from Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt.

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Annaby, M.H., Tharwat, M.M. A sinc-method computation for eigenvalues of Schrödinger operators with eigenparameter-dependent boundary conditions. Calcolo 54, 23–41 (2017). https://doi.org/10.1007/s10092-016-0174-3

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  • DOI: https://doi.org/10.1007/s10092-016-0174-3

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