Numerical Algorithms

, Volume 80, Issue 3, pp 795–817 | Cite as

Sinc-regularized techniques to compute eigenvalues of schrödinger operators on \(L^{2}(I)\oplus \mathbb {C}^{2}\)

  • M. H. AnnabyEmail author
  • M. M. Tharwat
Original Paper


We introduced two different sinc-regularized techniques to compute eigenvalues of Schrödinger operators defined on the Hilbert space \(L^{2}(0,1)\oplus \mathbb {C}^{2}\) because of the appearance of eigenvalue parameter in the boundary conditions. Both techniques improve significantly the error estimates resulting from using the classical sampling method. Furthermore, it treats completely the obstacle that, for some potentials, integrals resulting from applying the sinc method cannot be explicitly computed. We derive a rigorous error analysis that takes into account both truncation and perturbation errors. Results are exhibited numerically and graphically with comparisons.


Sinc-Gaussian method Sinc method Eigenvalue problem with eigenparameter in the boundary conditions Truncation and amplitude errors 

Mathematics Subject Classification (2010)

34L16 94A20 65L15 


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  1. 1.
    Annaby, M.H., Asharabi, R.M.: Computing eigenvalues of boundary-value problems using sinc-Gaussian method. Sampl. Theory Signal Image Process. 7, 293–312 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Annaby, M.H., Tharwat, M.M.: On computing eigenvalues of second-order linear pencils. IMA J. Numer. Anal. 27, 366–380 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Annaby, M.H., Tharwat, M.M.: Sinc-based computations of eigenvalues of Dirac systems. BIT Numer. Math. 47, 699–713 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Annaby, M.H., Tharwat, M.M.: A sinc-method computation for eigenvalues of Schrödinger o with eigenparameter-dependent boundary conditions. Calcolo 54, 23–41 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Binding, P.A., Browne, P.J., Watson, B.A.: Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. I Proc. Edin. Math. Soc. 45, 631–645 (2002)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Binding, P.A., Browne, P.J., Watson, B.A.: Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. II J. Comput. Appl. Math. 148, 147–168 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Butzer, P.L., Schmeisser, G., Stens, R.L. In: Marvasti, F. (ed.) : An introduction to sampling analysis, pp 17–121. Kluwer, New York (2001)Google Scholar
  9. 9.
    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)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering Theory, 2nd edn. Springer–Verlag, Berlin (1989)CrossRefzbMATHGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eastham, M.S.P.: Theory of Ordinary Differential Equations. Van Nostrand Reinhold, London (1970)zbMATHGoogle Scholar
  13. 13.
    Eggert, N., Jarratt, M., Lund, J.: Sinc function computation of the eigenvalue of Sturm-Liouville problems. J. Comput. Phys. 69, 209–229 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fulton, C.T.: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edin, Sec. A: Math. 77, 293–308 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gaskell, R.E.: A problem in heat conduction and an expansion theorem. Amer. J. Math. 64, 447–455 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Non-Self-Adjoint Operators in Hilbert Space AMS Translations of Mathematical Monographs, vol. 18. AMS, Providence (1969)Google Scholar
  17. 17.
    Jagerman, D.: Bounds for truncation error of the sampling expansion. SIAM J. Appl. Math. 14, 714–723 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kemp, R.R.D.: Operators on \(L^{2}\mathbb {C}^{r}\). Can. J. Math. 39, 33–53 (1987)CrossRefGoogle Scholar
  19. 19.
    Kemp, R.R.D., Lee, S.J.: Finite dimensional perturbations of differential expressions. Can J. Math. 28, 1082–1104 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kotel’nikov, V.: On the carrying capacity of the “ether” and wire in telecommunications, material for the first All-Union Conference on Questions of Communications. Izd. Red. Upr. Svyazi RKKA, Moscow, Russian 55, 55–64 (1933)Google Scholar
  21. 21.
    Langer, R.E.: A problem in diffusion or in the flow of heat for a solid in contact with a fluid. Tohoku Math. J 35, 360–375 (1932)Google Scholar
  22. 22.
    Lund, J., Bowers, K.: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia (1992)CrossRefzbMATHGoogle Scholar
  23. 23.
    Mennicken, R., Möller, M.: Non-Self-Adjoint Boundary Eigenvalue Problems, North-Holland Mathematics studies, vol. 192. Elsevier, Amsterdam (2003)Google Scholar
  24. 24.
    Naimark, M.A.: Linear Differential Operators. George Harrap, London (1967)zbMATHGoogle Scholar
  25. 25.
    Qian, L.: On the regularized Whittaker-Kotel’nikov-Shannon sampling formula. Proc. Amer. Math. Soc. 131, 1169–1176 (2002)CrossRefzbMATHGoogle Scholar
  26. 26.
    Qian, L., Creamer, D.B.: A modification of the sampling series with a Gaussian multiplie. Sampl. Theory Signal Image Process. 5, 1–20 (2006)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Qian, L., Creamer, D.B.: Localized sampling in the presence of noise. Appl. Math. Lett. 19, 351–355 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Schmeisser, G., Stenger, F.: Sinc approximation with a Gaussian multiplier. Sampl. Theory Signal Image Process. 6, 199–221 (2007)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Shannon, C.E.: Communications in the presence of noise. Proc. I.R.E. 37, 10–21 (1949)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer-Verlag, New York (1993)CrossRefzbMATHGoogle Scholar
  31. 31.
    Stenger, F.: Handbook of Sinc Numerical Methods. CRC Press, Boca Raton (2011)zbMATHGoogle Scholar
  32. 32.
    Tretter, C.: Spectral problems for systems of differential equations y + A 0 y = A 1 y with λ-polynomial boundary conditions. Math. Nach. 214, 129–172 (2000)CrossRefGoogle Scholar
  33. 33.
    Tretter, C.: Boundary eigenvalue problems for differential equations N η = P η with λ-polynomial boundary conditions. J. Diff. Eq. 170, 408–471 (2001)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Whittaker, E.T.: On the functions which are represented by the expansion of the interpolation theory. Proc. Roy. Soc. Edinburgh Sect. A 35, 181–194 (1915)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceCairo UniversityGizaEgypt
  2. 2.Department of Mathematics, Faculty of ScienceBeni-Suef UniversityBeni-SuefEgypt

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