Open Access

On nonsymmetric rank one singular perturbations of selfadjoint operators


We consider nonsymmetric rank one singular perturbations of a selfadjoint operator, i.e., an expression of the form $\tilde A=A+\alpha\left\langle\cdot,\omega_1\right\rangle\omega_2$, $\omega_1\not=\omega_2$, $\alpha\in{\mathbb C}$, in a general case $\omega_1,\omega_2\in{\mathcal H}_{-2}$. Using a constructive description of the perturbed operator $\tilde A$, we investigate some spectral and approximations properties of $\tilde A$. The wave operators corresponding to the couple $A$, $\tilde A$ and a series of examples are also presented.

Key words: Singular perturbation, nonsymmetric perturbations, eigenvalue problem, M. Krein's formula.

Full Text

Article Information

TitleOn nonsymmetric rank one singular perturbations of selfadjoint operators
SourceMethods Funct. Anal. Topology, Vol. 22 (2016), no. 2, 137-151
MathSciNet MR3522856
zbMATH 06665384
MilestonesReceived 22/09/2015; Revised 23/02/2016
CopyrightThe Author(s) 2016 (CC BY-SA)

Authors Information

Mykola Dudkin
National Technical University of Ukraine "Kyiv Polytechnic Institute", 37 Prospect Peremogy, Kyiv, 03056, Ukraine

Tetiana Vdovenko
National Technical University of Ukraine "Kyiv Polytechnic Institute", 37 Prospect Peremogy, Kyiv, 03056, Ukraine

Citation Example

Mykola Dudkin and Tetiana Vdovenko, On nonsymmetric rank one singular perturbations of selfadjoint operators, Methods Funct. Anal. Topology 22 (2016), no. 2, 137-151.


@article {MFAT841,
    AUTHOR = {Dudkin, Mykola and Vdovenko, Tetiana},
     TITLE = {On nonsymmetric rank one singular perturbations of selfadjoint operators},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {2},
     PAGES = {137-151},
      ISSN = {1029-3531},
  MRNUMBER = {MR3522856},
 ZBLNUMBER = {06665384},
       URL = {},

Google Scholar Metrics

Citing articles in Google Scholar
Similar articles in Google Scholar

Export article

Save to Mendeley


  1. S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden, Solvable models in quantum mechanics, AMS Chelsea Publishing, Providence, RI, 2005.  MathSciNet
  2. Sergio Albeverio, Mykola Dudkin, and Volodymyr Koshmanenko, Rank-one singular perturbations with a dual pair of eigenvalues, Lett. Math. Phys. 63 (2003), no. 3, 219-228.  MathSciNet CrossRef
  3. S. Albeverio and P. Kurasov, Singular perturbations of differential operators, London Mathematical Society Lecture Note Series, vol. 271, Cambridge University Press, Cambridge, 2000.  MathSciNet CrossRef
  4. S. Albeverio, V. Koshmanenko, P. Kurasov, and L. Nizhnik, On approximations of rank one $\scr H_ -2$-perturbations, Proc. Amer. Math. Soc. 131 (2003), no. 5, 1443-1452.  MathSciNet CrossRef
  5. Sergio Albeverio, Sergei Kuzhel, and Leonid P. Nizhnik, On the perturbation theory of self-adjoint operators, Tokyo J. Math. 31 (2008), no. 2, 273-292.  MathSciNet CrossRef
  6. Sergio Albeverio and Leonid Nizhnik, Schrodinger operators with nonlocal potentials, Methods Funct. Anal. Topology 19 (2013), no. 3, 199-210.  MathSciNet MFAT Article
  7. Yurij M. Berezansky and Johannes Brasche, Generalized selfadjoint operators and their singular perturbations, Methods Funct. Anal. Topology 8 (2002), no. 4, 1-14.  MathSciNet MFAT Article
  8. Yu. M. Berezansky, Z. G. Sheftel, G. F. Us, Functional Analysis, Vols. 1, 2, Birkhauser Verlag, Basel-Boston-Berlin, 1996. (Russian edition: Vyshcha shkola, Kiev, 1990)
  9. M. E. Dudkin, Invariant symmetric restrictions of a selfadjoint operator. II, Ukrainian Math. J. 50 (1998), no. 6, 888-900.  MathSciNet CrossRef
  10. M. E. Dudkin, Singularly perturbed normal operators, Ukrainian Mat. J. 51 (1999), no. 8, 1177-1187.  MathSciNet CrossRef
  11. Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  MathSciNet
  12. Tosio Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1966), no. 2, 258-279.  MathSciNet CrossRef
  13. Volodymyr Koshmanenko, Singular quadratic forms in perturbation theory, Mathematics and its Applications, vol. 474, Kluwer Academic Publishers, Dordrecht, 1999.  MathSciNet CrossRef
  14. V. Koshmanenko, Singularly perturbed operators, Mathematical results in quantum mechanics (Blossin, 1993), Oper. Theory Adv. Appl., vol. 70, Birkhauser, Basel, 1994, pp. 347-351.  MathSciNet CrossRef
  15. V. B. Lidskii, A non-selfadjoint operator of Sturm-Liouville type with a discrete spectrum, Trudy Mosk. Mat. Obsh. 9 (1960), 45-79.  MathSciNet
  16. M. M. Malamud and V. I. Mogilevskii, Krein type formula for canonical resolvents of dual pairs of linear relations, Methods Funct. Anal. Topology 8 (2002), no. 4, 72-100.  MathSciNet MFAT Article
  17. Leonid Nizhnik, Inverse nonlocal Sturm-Liouville problem, Inverse Problems 26 (2010), no. 12, 125006, 9 pp.  MathSciNet CrossRef
  18. Leonid Nizhnik, Inverse spectral nonlocal problem for the first order ordinary differential equation, Tamkang J. Math. 42 (2011), no. 3, 385-394.  MathSciNet CrossRef
  19. M. I. Vishik, On general boundary-value problems for elliptic differential equation, Trudy Mosk. Mat. Obsh. 1 (1952), 187-246. (Russian)
  20. T. Vdovenko and M. Dudkin, Singular rank one nonsymmetric perturbations of a selfadjoint operator, Proceedings of Institute of Mathematics of NAS of Ukraine, 12 (2015), no. 1, 57-73. (Ukrainian)

All Issues