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.
Mykola Dudkin and Tetiana Vdovenko, On nonsymmetric rank one singular perturbations of selfadjoint operators, Methods Funct. Anal. Topology 22
(2016), no. 2, 137-151.
BibTex
@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 = {http://mfat.imath.kiev.ua/article/?id=841},
}
References
S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden, Solvable models in quantum mechanics, AMS Chelsea Publishing, Providence, RI, 2005. MathSciNet
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. MathSciNetCrossRef
S. Albeverio and P. Kurasov, Singular perturbations of differential operators, London Mathematical Society Lecture Note Series, vol. 271, Cambridge University Press, Cambridge, 2000. MathSciNetCrossRef
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. MathSciNetCrossRef
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. MathSciNetCrossRef
Sergio Albeverio and Leonid Nizhnik, Schrodinger operators with nonlocal potentials, Methods Funct. Anal. Topology 19 (2013), no. 3, 199-210. MathSciNetMFAT Article
Yurij M. Berezansky and Johannes Brasche, Generalized selfadjoint operators and their singular perturbations, Methods Funct. Anal. Topology 8 (2002), no. 4, 1-14. MathSciNetMFAT Article
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)
M. E. Dudkin, Invariant symmetric restrictions of a selfadjoint operator. II, Ukrainian Math. J. 50 (1998), no. 6, 888-900. MathSciNetCrossRef
M. E. Dudkin, Singularly perturbed normal operators, Ukrainian Mat. J. 51 (1999), no. 8, 1177-1187. MathSciNetCrossRef
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MathSciNet
Tosio Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1966), no. 2, 258-279. MathSciNetCrossRef
Volodymyr Koshmanenko, Singular quadratic forms in perturbation theory, Mathematics and its Applications, vol. 474, Kluwer Academic Publishers, Dordrecht, 1999. MathSciNetCrossRef
V. Koshmanenko, Singularly perturbed operators, Mathematical results in quantum mechanics (Blossin, 1993), Oper. Theory Adv. Appl., vol. 70, Birkhauser, Basel, 1994, pp. 347-351. MathSciNetCrossRef
V. B. Lidskii, A non-selfadjoint operator of Sturm-Liouville type with a discrete spectrum, Trudy Mosk. Mat. Obsh. 9 (1960), 45-79. MathSciNet
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. MathSciNetMFAT Article
Leonid Nizhnik, Inverse nonlocal Sturm-Liouville problem, Inverse Problems 26 (2010), no. 12, 125006, 9 pp. MathSciNetCrossRef
Leonid Nizhnik, Inverse spectral nonlocal problem for the first order ordinary differential equation, Tamkang J. Math. 42 (2011), no. 3, 385-394. MathSciNetCrossRef
M. I. Vishik, On general boundary-value problems for elliptic differential equation, Trudy Mosk. Mat. Obsh. 1 (1952), 187-246. (Russian)
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)