Advertisement

Journal of Mathematical Chemistry

, Volume 55, Issue 6, pp 1253–1277 | Cite as

Zero interval limit perturbation expansion for the spectral entities of Hilbert–Schmidt operators combined with most dominant spectral component extraction: formulation and certain technicalities

Original Paper

Abstract

A perturbation-expansion-at-zero-interval-limit based numeric al algorithm to calculate the eigenpairs of Hilbert–Schmidt integral operators having symmetric kernels is developed in the present work. We have developed the perturbation expansion only for the most dominant eigenvalue and relevant eigenfunction. The less important eigenpairs have been determined by using the most dominant spectral component extraction recursively over the kernel restrictions. The main lines of the formulation and certain related technicalities are presented here. The confirmation of the presented theory via certain illustrative implementations and the convergence discussion for the obtained perturbation series as well as the numerical comparison with some mostly considered methods are given in the next companion of this paper.

Keywords

Hilbert–Schmidt integral operators Perturbation expansions Eigenvalues Eigenfunctions Singular value decomposition for continuous functions Most dominant spectral component extraction 

Mathematics Subject Classification

45B05 45F10 45P05 47A55 65D15 

References

  1. 1.
    H.H. Gan, B.C. Eu, J. Chem. Phys. (1993). doi: 10.1063/1.466106 Google Scholar
  2. 2.
    E. Cancés, B. Mennucci, J. Math. Chem. (1998). doi: 10.1023/A:1019133611148 Google Scholar
  3. 3.
    T. Buchukuri, O. Chkadua, D. Natroshvili, Integral Equ. Oper. Theory (2009). doi: 10.1007/s00020-009-1694-x Google Scholar
  4. 4.
    S. Arnrich, P. Bräuer, G. Kalies, J. Math. Chem. (2015). doi: 10.1007/s10910-015-0531-5 Google Scholar
  5. 5.
    A. Townsend, L.N. Trefethen, Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. (2014) doi: 10.1098/rspa.2014.0585
  6. 6.
    S. Tuna, M. Demiralp, in AIP Conference Proceedings, vol. 1702 (2015), p. 170009. doi: 10.1063/1.4938944
  7. 7.
    B. Tunga, M. Demiralp, J. Math. Chem. (2010). doi: 10.1007/s10910-010-9714-2 Google Scholar
  8. 8.
    M.A. Tunga, M. Demiralp, J. Math. Chem. (2011). doi: 10.1063/1.3637819 Google Scholar
  9. 9.
    S. Tuna, B. Tunga, J. Math. Chem. (2012). doi: 10.1007/s10910-013-0179-y Google Scholar
  10. 10.
    M.A. Tunga, M. Demiralp, J. Math. Chem. (2013). doi: 10.1007/s10910-013-0228-6 Google Scholar
  11. 11.
    M.A. Tunga, Int. J. Comput. Math. (2014). doi: 10.1080/00207160.2014.941825 Google Scholar
  12. 12.
    E. Korkmaz Özay, M. Demiralp, J. Math. Chem. (2014). doi: 10.1007/s10910-014-0396-z
  13. 13.
    M.A. Tunga, Int. J. Comput. Math. (2015). doi: 10.1080/00207160.2014.941825 Google Scholar
  14. 14.
    L. Debnath, P. Mikusiński, Introduction to Hilbert Spaces with Applications, 3rd edn. (Elsevier, Amsterdam, 2005)Google Scholar
  15. 15.
    F. Chatelin, Spectral Approximation of Linear Operators (SIAM, Philedelphia, 2011)CrossRefGoogle Scholar
  16. 16.
    F.G. Tricomi, Integral Equations (Interscience Publishers, New York, 1957)Google Scholar
  17. 17.
    L.N. Trefethen, D. Bau, Numerical Linear Algebra (SIAM, Philadelphia, 1997)CrossRefGoogle Scholar
  18. 18.
    C. Corduneanu, Math. Syst. Theory (1967). doi: 10.1007/BF01705524 Google Scholar
  19. 19.
    T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1995)Google Scholar
  20. 20.
    E.J. Hinch, Perturbation Methods (Cambridge University Press, Cambridge, 1991)CrossRefGoogle Scholar
  21. 21.
    M.H. Holmes, Introduction to Perturbation Methods (Springer, New York, 1995)CrossRefGoogle Scholar
  22. 22.
    I.M. Sobol, Math. Model. Comput. Exp. 1, 407–414 (1993)Google Scholar
  23. 23.
    H. Rabitz, Ö.F. Alış, J. Shorter, K. Shim, Comput. Phys. Commun. (1999). doi: 10.1016/S0010-4655(98)00152-0 Google Scholar
  24. 24.
    Ö.F. Alış, H. Rabitz, J. Math. Chem. (2001). doi: 10.1023/A:1010979129659 Google Scholar
  25. 25.
    M.A. Tunga, M. Demiralp, Appl. Math. Comput. (2005). doi: 10.1016/j.amc.2004.06.056 Google Scholar
  26. 26.
    M.A. Tunga, M. Demiralp, Int. J. Comput. Math. (2008). doi: 10.1080/00207160701576095 Google Scholar
  27. 27.
    W.W. Bell, Special Functions for Scientists and Engineers (Dover, New York, 2004)Google Scholar
  28. 28.
    S. Tuna, M. Demiralp, J. Math. Chem. (2017). doi: 10.1007/s10910-017-0740-1
  29. 29.
    W. Oevel, F. Postel, S. Wehmeier, J. Gerhard, The MuPAD Tutorial (Springer, Berlin, 2000)CrossRefGoogle Scholar
  30. 30.
    E. Korkmaz Özay, M. Demiralp, J. Math. Chem. (2012). doi: 10.1007/s10910-012-0018-6
  31. 31.
    E. Korkmaz Özay, M. Demiralp, Combined small scale enhanced multivariance product representation, in Proceedings of the International Conference on Applied Computer Science (2010), p. 350356Google Scholar
  32. 32.
    S. Tuna, M. Demiralp, Mathematics (2017). doi: 10.3390/math5010002 Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Informatics Instituteİstanbul Technical UniversityMaslakTurkey

Personalised recommendations