Differential Equations

, Volume 55, Issue 10, pp 1328–1335 | Cite as

Changes in a Finite Part of the Spectrum of the Laplace Operator under Delta-Like Perturbations

  • B. E. KanguzhinEmail author
Partial Differential Equations


We study the spectrum of the Laplace operator in a bounded simply connected domain with the zero Dirichlet condition on the boundary under delta-like perturbations of the operator at an interior point of the domain. We determine the maximal operator for the perturbations and single out a class of invertible restrictions of this operator whose spectra differ from the spectrum of the original operator by a finite (possibly, empty) set. These results can be viewed as transferring some of H. Hochstadt’s results for Sturm-Liouville operators to Laplace operators.


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This work was supported in part by the Ministry of Education and Science of the Republic of Kazakhstan, project no. AP05131292.


  1. 1.
    Mizohata, S., The Theory of Partial Differential Equations, Cambridge: Cambridge Univ., 1973. Translated under the title Teoriya uravnenii s chastnymi proizvodnymi, Moscow: Mir, 1977.zbMATHGoogle Scholar
  2. 2.
    Kanguzhin, B.E., Weinstein criteria and regularized traces in the case of transverse vibrations of an elastic string with springs, Differ. Equations, 2018, vol. 54, no. 1, pp. 7–12.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Crum, M.M., Associated Sturm-Liouville systems, Quart. S. Math., 1955, vol. 6, pp. 121–127.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hochstadt, H., The inverse Sturm-Liouville problem, Comm. Pure Appl. Math., 1973, vol. 26, pp. 715–729.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Levitan, B.M., On the definition of the Sturm-Liouville operator based on one and two spectra, Izv. Akad. Nauk SSSR Ser. Mat., 1978, vol. 42, no. 1, pp. 185–199.MathSciNetGoogle Scholar
  6. 6.
    Hochstadt, H., On some inverse problems in matrix theory, Arch. Math., 1967, vol. 18, no. 2, pp. 201–207.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hochstadt, H. and Lieberman, B., An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., 1978, vol. 34, no. 4, pp. 676–680.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Savchuk, A.M. and Shkalikov, A.A., Sturm-Liouville operators with singular potentials, Math. Notes, 1999, vol. 66, no. 6, pp. 741–753.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kanguzhin, B.E. and Aniyarov, A.A., Well-posed problems for the Laplace operator in a punctured disk, Math. Notes, 2011, vol. 89, no. 6, pp. 819–829.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kanguzhin, B.E. and Tokmagambetov, N.E., Resolvents of well-posed problems for finite-rank perturbations of the polyharmonic operator in a punctured domain, Sib. Math. J., 2016, vol. 57, no. 2, pp. 265–273.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sadovnichii, V.A. and Lyubishkin, V.A., Finite-dimensional perturbations of discrete operators and trace formulas, Funkts. Anal. Ego Prilozh., 1986, vol. 20, no. 3, pp. 55–65.MathSciNetGoogle Scholar
  12. 12.
    Birman, M.Sh. and Yafaev, D.R., Spectral shift function. Works of M.G. Krein and their further development, Algebra i Anal., 1992, vol. 4, no. 5, pp. 1–42.Google Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Al-Farabi Kazakh National UniversityAlmatyKazakhstan
  2. 2.Institute of Mathematics and Mathematical ModelingAlmatyKazakhstan

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