Advertisement

Eigenvalue Bounds for Self-Adjoint Eigenvalue Problems

  • Mitsuhiro T. Nakao
  • Michael Plum
  • Yoshitaka Watanabe
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 53)

Abstract

Eigenvalue problems Au = λu with a self-adjoint operator A are ubiquitous in mathematical analysis and mathematical physics. A particularly rich field of application is formed by linear differential expressions which can be realized operator-theoretically by self-adjoint operators. Often such eigenvalue problems arise from wave- or Schrödinger-type equations after separation of the time variable, i.e. by a standing-wave ansatz. Possibly the most important physical application is quantum physics, but also other fields like electro-dynamics (including optics) or statistical mechanics are governed by partial differential operators and related eigenvalue problems.

References

  1. 3.
    Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Computer Science and Applied Mathematics. Academic [Harcourt Brace Jovanovich, Publishers], New York (1983). Translated from the German by Jon RokneCrossRefGoogle Scholar
  2. 19.
    Aronszajn, N.: Approximation eigenvalues of completely continuous symmetric ooperators. In: Proceedings of the Spectral Theory and Differential Problems, pp. 179–202 (1951)Google Scholar
  3. 23.
    Bazley, N.W., Fox, D.W.: A procedure for estimating eigenvalues. J. Math. Phys. 3, 469–471 (1962)MathSciNetCrossRefGoogle Scholar
  4. 24.
    Bazley, N.W., Fox, D.W.: Comparison operators for lower bounds to eigenvalues. J. Reine Angew. Math. 223, 142–149 (1966)MathSciNetzbMATHGoogle Scholar
  5. 25.
    Beattie, C.: An extension of Aronszajn’s rule: slicing the spectrum for intermediate problems. SIAM J. Numer. Anal. 24(4), 828–843 (1987)MathSciNetCrossRefGoogle Scholar
  6. 26.
    Beattie, C., Goerisch, F.: Methods for computing lower bounds to eigenvalues of self-adjoint operators. Numer. Math. 72(2), 143–172 (1995)MathSciNetCrossRefGoogle Scholar
  7. 27.
    Behnke, H.: Inclusion of eigenvalues of general eigenvalue problems for matrices. In: Scientific Computation with Automatic Result Verification (Karlsruhe, 1987). Volume 6 of Computing Supplementum, pp. 69–78. Springer, Vienna (1988)Google Scholar
  8. 28.
    Behnke, H., Goerisch, F.: Inclusions for eigenvalues of selfadjoint problems. In: Topics in Validated Computations, Oldenburg, 1993. Volume 5 of Studies in Computational Mathematics, pp. 277–322. North-Holland, Amsterdam (1994)Google Scholar
  9. 29.
    Behnke, H., Mertins, U., Plum, M., Wieners, C.: Eigenvalue inclusions via domain decomposition. Proc. R. Soc. Lond. A 456, 2717–2730 (2000)CrossRefGoogle Scholar
  10. 30.
    Berkowitz, J.: On the discreteness of spectra of singular Sturm-Liouville problems. Commun. Pure Appl. Math. 12, 523–542 (1959)MathSciNetCrossRefGoogle Scholar
  11. 31.
    Birkhoff, G., de Boor, C., Swartz, B., Wendroff, B.: Rayleigh-Ritz approximation by piecewise cubic polynomials. SIAM J. Numer. Anal. 3, 188–203 (1966)MathSciNetCrossRefGoogle Scholar
  12. 32.
    Birman, M.S., Solomjak, M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Mathematics and Its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht (1987). Translated from the 1980 Russian original by S. Khrushchëv and V. PellerGoogle Scholar
  13. 48.
    Chatelin, F.: Spectral Approximation of Linear Operators. Volume 65 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2011). With a foreword by P. Henrici, With solutions to exercises by Mario Ahués, Reprint of the 1983 original [MR0716134]Google Scholar
  14. 62.
    Davies, E.B.: A hierarchical method for obtaining eigenvalue enclosures. Math. Comput. 69(232), 1435–1455 (2000)MathSciNetCrossRefGoogle Scholar
  15. 63.
    Davies, E.B., Parnovski, L.: Trapped modes in acoustic waveguides. Quart. J. Mech. Appl. Math. 51(3), 477–492 (1998)MathSciNetCrossRefGoogle Scholar
  16. 64.
    Davies, E.B., Plum, M.: Spectral pollution. IMA J. Numer. Anal. 24(3), 417–438 (2004)MathSciNetCrossRefGoogle Scholar
  17. 74.
    Evans, D.V., Linton, C.M.: Trapped modes in open channels. J. Fluid Mech. 225, 153–175 (1991)MathSciNetCrossRefGoogle Scholar
  18. 89.
    Goerisch, F.: Ein Stufenverfahren zur Berechnung von Eigenwertschranken. In Numerische Behandlung von Eigenwertaufgaben (eds. J. Albrecht, L. Collatz, W. Velte). Internationale Schriftenreiche Numerische Mathematik, 83, 104–114 (1987)MathSciNetGoogle Scholar
  19. 98.
    Hammer, R., Hocks, M., Kulisch, U., Ratz, D.: Numerical Toolbox for Verified Computing. I. Volume 21 of Springer Series in Computational Mathematics. Springer, Berlin (1993). Basic numerical problems, theory, algorithms, and Pascal-XSC programs, with separately available computer disksGoogle Scholar
  20. 110.
    Kato, T.: On the upper and lower bounds of eigenvalues. J. Phys. Soc. Japan 4, 334–339 (1949)MathSciNetCrossRefGoogle Scholar
  21. 116.
    Kikuchi, F., Liu, X.: Determination of the Babuska-Aziz constant for the linear triangular finite element. Japan J. Indust. Appl. Math. 23(1), 75–82 (2006)MathSciNetCrossRefGoogle Scholar
  22. 117.
    Kikuchi, F., Liu, X.: Estimation of interpolation error constants for the P 0 and P 1 triangular finite elements. Comput. Methods Appl. Mech. Eng. 196(37–40), 3750–3758 (2007)MathSciNetCrossRefGoogle Scholar
  23. 123.
    Klatte, R., Kulisch, U., Lawo, C., Rausch, M., Wiethoff, A.: C-XSC-A C++ Library for Extended Scientific Computing. Springer, Berlin (1993)zbMATHGoogle Scholar
  24. 137.
    Lehmann, N.J.: Optimale Eigenwerteinschließungen. Numer. Math. 5, 246–272 (1963)CrossRefGoogle Scholar
  25. 143.
    Liu, X.: A framework of verified eigenvalue bounds for self-adjoint differential operators. Appl. Math. Comput. 267, 341–355 (2015)MathSciNetzbMATHGoogle Scholar
  26. 145.
    Liu, X., Oishi, S.: Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape. SIAM J. Numer. Anal. 51(3), 1634–1654 (2013)MathSciNetCrossRefGoogle Scholar
  27. 148.
    Maehly, H.J.: Ein neues Variationsverfahren zur genäherten Berechnung der Eigenwerte hermitescher Operatoren. Helvetica Phys. Acta 25, 547–568 (1952)MathSciNetzbMATHGoogle Scholar
  28. 181.
    Nagatou, K., Plum, M., Nakao, M.T.: Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468(2138), 545–562 (2012)MathSciNetCrossRefGoogle Scholar
  29. 227.
    Plum, M.: Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method. Z. Angew. Math. Phys. 41(2), 205–226 (1990)MathSciNetCrossRefGoogle Scholar
  30. 229.
    Plum, M.: Bounds for eigenvalues of second-order elliptic differential operators. Z. Angew. Math. Phys. 42(6), 848–863 (1991)MathSciNetCrossRefGoogle Scholar
  31. 233.
    Plum, M.: Guaranteed numerical bounds for eigenvalues. In: Spectral Theory and Computational Methods of Sturm-Liouville Problems, Knoxville, 1996. Volume 191 of Lecture Notes in Pure and Applied Mathematics, pp. 313–332. Dekker, New York (1997)Google Scholar
  32. 239.
    Rektorys, K.: Variational Methods in Mathematics, Science and Engineering, 2nd edn. D. Reidel Publishing Co., Dordrecht/Boston (1980). Translated from the Czech by Michael BaschzbMATHGoogle Scholar
  33. 241.
    Riesz, F., Sz.-Nagy, B.: Leçons d’analyse fonctionnelle. Gauthier-Villars, Editeur-Imprimeur-Libraire, Paris; Akadémiai Kiadó, Budapest (1965). Quatrième édition. Académie des Sciences de HongrieGoogle Scholar
  34. 244.
    Rump, S.M.: Intlab-interval laboratory, a matlab toolbox for verified computations, version 4.2.1. (2002)Google Scholar
  35. 259.
    Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Volume 265 of Graduate Texts in Mathematics. Springer, Dordrecht (2012)Google Scholar
  36. 295.
    Weinberger, H.F.: Variational Methods for Eigenvalue Approximation. Society for Industrial and Applied Mathematics, Philadelphia (1974). Based on a Series of Lectures Presented at the NSF-CBMS Regional Conference on Approximation of Eigenvalues of Differential Operators, Vanderbilt University, Nashville, 26–30 June 1972, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 15Google Scholar
  37. 296.
    Weinstein, A.: On the Sturm-Liouville theory and the eigenvalues of intermediate problems. Numer. Math. 5, 238–245 (1963)MathSciNetCrossRefGoogle Scholar
  38. 297.
    Weinstein, A., Stenger, W.: Methods of Intermediate Problems for Eigenvalues. Theory and Ramifications. Volume 89 of Mathematics in Science and Engineering. Academic Press, New York/London (1972)Google Scholar
  39. 298.
    Wieners, C.: Bounds for the N lowest eigenvalues of fourth-order boundary value problems. Computing 59(1), 29–41 (1997)MathSciNetCrossRefGoogle Scholar
  40. 313.
    You, C., Liu, X., Xie, H., Plum, M.: High-precision guaranteed eigenvalue bounds for the Steklov eigenvalue problem (In preparation)Google Scholar
  41. 323.
    Zimmermann, S., Mertins, U.: Variational bounds to eigenvalues of self-adjoint eigenvalue problems with arbitrary spectrum. Z. Anal. Anwendungen 14(2), 327–345 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Mitsuhiro T. Nakao
    • 1
  • Michael Plum
    • 2
  • Yoshitaka Watanabe
    • 3
  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Faculty of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany
  3. 3.Research Institute for Information TechnologyKyushu UniversityFukuokaJapan

Personalised recommendations