Dual pair of eigenvalues in rank one singular nonsymmetric
perturbations

T. I. Vdovenko; M. E. Dudkin

doi:10.15330/ms.48.1.156-164

In the separable Hilbert space, we discuss the eigenvalue problem for a rank one singular nonselfadjoint perturbation of a selfadjoint operator $A$, by nonsymmetric potential ($\delta_1\not=\delta_2$) in the form $\tilde A=A+\alpha\left\langle\cdot,\delta_1\right\rangle\delta_2$. We give the constructive description of such sort operator $\tilde A$ which possess two new points in the point spectrum in case of weakly singular perturbations.

1. S. Albeverio, F. Gesztesy, R. Hegh-Krohn, H. Holden, Solvable models in quantum mechanics. Second edition, With an appendix by Pavel Exner, AMS Chelsea Publishing, Providence, RI, 2005.

2. S. Albeverio, M. Dudkin, A. Konstantinov, V. Koshmanenko, On the point spectrum of $H_{-2}$-singular perturbations, Math. Nachr., 280 (2007), №1.2, 20-27.

3. S. Albeverio, M. Dudkin, V. Koshmanenko, Dual pair of eigenvalues, Letters in Math. Phis., 63 (2003), 219-228.

4. S. Albeverio, R. Hryniv, L. Nizhnik Inverse spectral problems for nonlocal Sturm-Liouville operators, Inverse Problems, 23 (2007), 523-535.

5. S. Albeverio, P. Kurasov, Singular perturbations of differential operators. Solvable SchrЃNodinger type operators, London Mathematical Society Lecture Note Series, 271, Cambridge University Press, Cambridge, 2000.

6. Y. Berezansky, J. Brasche, Generalized selfadjoint operators and their singular perturbations, Methods Funct. Anal. Topology, 7 (2001), №3, 54-66.

7. M.E. Dudkin, T.I. Vdovenko, Syngular rank one non-symmetric perturbations of a self-adjoint operator. in The spectral theorey of operators and sets of operators, Proceedings of Institute of Mathematics of NAS of Ukraine, 12 (2015), №1, 57-73. (in Ukrainian)

8. M.E. Dudkin, Singularly perturbed rank one normal operators and its applications, Preprint, Institute of Mathematics NAS of Ukraine, 2008, 38 p.

9. M.E. Dudkin, T.I. Vdovenko, Rank one strong singular perturbation by non-symmetric potential, Research Bulletin of National Technical Univ. of Ukraine KPI, 96 (2014), 13-17.

10. M.E. Dudkin, V.D. Koshmanenko, The point spectrum of self-adjoint operators that appears under singular perturbations of finite rank, Ukrainian Math. J., 55 (2003), №9, 1532-1541.

11. T. Kato, Wave operators and similarity for some non self-adjoint operators, Math. Annalen, 162 (1966), 258-279.

12. T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition. Classics in Mathematics, Springer-Verlag, Berlin, 1995.

13. T. Kato, J.B. McLeod, The functional-differential equation $y'(x)=ay(\lambda x)+by(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937.

14. V. Koshmanenko, Singular quadratic forms in perturbation theory, Translated from the 1993 Russian original by P. V. Malyshev and D. V. Malyshev, Mathematics and its Applications, 474., Kluwer Academic Publishers, Dordrecht, 1999.

15. V.B. Lidskii, The non self-adjoint operator of Sturm-Liouville type with discrete spectrum, Trudy Moskow. Mat. Obshchestva, 9 (1960), 45-79. (in Russian)

16. M.M. Malamud, V.I. Mogilevskii, Krein type formula for canonical resolvents of dual pairs of linear relations, Methods Funct. Anal. Topology, 8 (2002), №4, 72-100.

17. A.D. Myshkis, General theory of differential equations with retarded arguments, Uspekhi Mat. Nauk, 4 (1949), №5(33), 99-141. (in Russian)

18. L. Nizhnik, Inverse nonlocal Sturm-Liouville problem, Inverse problems, 26, (2010), 9 p.

19. L. Nizhnik, Inverse spectral nonlocal problem for the first order ordinary differential equation, Tamkang Journal of Mathematics, 42 (2011), №3, 385-394.

20. M. Reed, B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York-San Francisco-London, 1978.

21. T.I. Vdovenko, The couple of eigenvalues of non-symmetric rank one singularly perturbed operators, Naukovi zapysky NaUKMA, 178 (2016), 3-8. (in Ukrainian)

	Dual pair of eigenvalues in rank one singular nonsymmetric perturbations
Author	T. I. Vdovenko, M. E. Dudkin tanyavdovenko@meta.ua; dudkin@imath.kiev.ua National Technical University of Ukraine, Kyiv, Ukraine
Abstract	In the separable Hilbert space, we discuss the eigenvalue problem for a rank one singular nonselfadjoint perturbation of a selfadjoint operator $A$, by nonsymmetric potential ($\delta_1\not=\delta_2$) in the form $\tilde A=A+\alpha\left\langle\cdot,\delta_1\right\rangle\delta_2$. We give the constructive description of such sort operator $\tilde A$ which possess two new points in the point spectrum in case of weakly singular perturbations.
Keywords	rank one singular perturbation; eigenvalue problem; M. Krein’s formula; nonselfadjoint perturbation; deviating argument
DOI	doi:10.15330/ms.48.2.156-164
Reference	1. S. Albeverio, F. Gesztesy, R. Hegh-Krohn, H. Holden, Solvable models in quantum mechanics. Second edition, With an appendix by Pavel Exner, AMS Chelsea Publishing, Providence, RI, 2005. 2. S. Albeverio, M. Dudkin, A. Konstantinov, V. Koshmanenko, On the point spectrum of $H_{-2}$-singular perturbations, Math. Nachr., 280 (2007), №1.2, 20-27. 3. S. Albeverio, M. Dudkin, V. Koshmanenko, Dual pair of eigenvalues, Letters in Math. Phis., 63 (2003), 219-228. 4. S. Albeverio, R. Hryniv, L. Nizhnik Inverse spectral problems for nonlocal Sturm-Liouville operators, Inverse Problems, 23 (2007), 523-535. 5. S. Albeverio, P. Kurasov, Singular perturbations of differential operators. Solvable SchrЃNodinger type operators, London Mathematical Society Lecture Note Series, 271, Cambridge University Press, Cambridge, 2000. 6. Y. Berezansky, J. Brasche, Generalized selfadjoint operators and their singular perturbations, Methods Funct. Anal. Topology, 7 (2001), №3, 54-66. 7. M.E. Dudkin, T.I. Vdovenko, Syngular rank one non-symmetric perturbations of a self-adjoint operator. in The spectral theorey of operators and sets of operators, Proceedings of Institute of Mathematics of NAS of Ukraine, 12 (2015), №1, 57-73. (in Ukrainian) 8. M.E. Dudkin, Singularly perturbed rank one normal operators and its applications, Preprint, Institute of Mathematics NAS of Ukraine, 2008, 38 p. 9. M.E. Dudkin, T.I. Vdovenko, Rank one strong singular perturbation by non-symmetric potential, Research Bulletin of National Technical Univ. of Ukraine KPI, 96 (2014), 13-17. 10. M.E. Dudkin, V.D. Koshmanenko, The point spectrum of self-adjoint operators that appears under singular perturbations of finite rank, Ukrainian Math. J., 55 (2003), №9, 1532-1541. 11. T. Kato, Wave operators and similarity for some non self-adjoint operators, Math. Annalen, 162 (1966), 258-279. 12. T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition. Classics in Mathematics, Springer-Verlag, Berlin, 1995. 13. T. Kato, J.B. McLeod, The functional-differential equation $y'(x)=ay(\lambda x)+by(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937. 14. V. Koshmanenko, Singular quadratic forms in perturbation theory, Translated from the 1993 Russian original by P. V. Malyshev and D. V. Malyshev, Mathematics and its Applications, 474., Kluwer Academic Publishers, Dordrecht, 1999. 15. V.B. Lidskii, The non self-adjoint operator of Sturm-Liouville type with discrete spectrum, Trudy Moskow. Mat. Obshchestva, 9 (1960), 45-79. (in Russian) 16. M.M. Malamud, V.I. Mogilevskii, Krein type formula for canonical resolvents of dual pairs of linear relations, Methods Funct. Anal. Topology, 8 (2002), №4, 72-100. 17. A.D. Myshkis, General theory of differential equations with retarded arguments, Uspekhi Mat. Nauk, 4 (1949), №5(33), 99-141. (in Russian) 18. L. Nizhnik, Inverse nonlocal Sturm-Liouville problem, Inverse problems, 26, (2010), 9 p. 19. L. Nizhnik, Inverse spectral nonlocal problem for the first order ordinary differential equation, Tamkang Journal of Mathematics, 42 (2011), №3, 385-394. 20. M. Reed, B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York-San Francisco-London, 1978. 21. T.I. Vdovenko, The couple of eigenvalues of non-symmetric rank one singularly perturbed operators, Naukovi zapysky NaUKMA, 178 (2016), 3-8. (in Ukrainian)
Pages	156-164
Volume	48
Issue	2
Year	2017
Journal	Matematychni Studii
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Table of content of issue	html

Dual pair of eigenvalues in rank one singular nonsymmetric perturbations