Wave Operators of a Rank One Singular Perturbation by Nonsymmetric Potential

Тетяна Іванівна Вдовенко

Abstract


Background. We consider nonselfadjoint singular rank one perturbation of a self-adjoint operator by nonsymmetric potential, i.e. expression of the form

\[
\tilde A=A+\alpha\left\langle\cdot,\delta_1\right\rangle\delta_2,
\] 

where A is a selfadjoint semi-bounded operator in the separable Hilbert space \[{\mathcal H}.\] Our investigations consist in the fact that it is unknown whether \[\tilde{A}\] is a spectral operator for vectors \[\delta_1,\delta_2\in{\mathcal H}_{-1},~\delta_1\not=\delta_2\].

Objective. Purpose of the study is to establish the existence of wave operators in \[\tilde{A}\] provided “weak-weak” singular perturbation rank “one-one”

\[{\rm dim}({\mathcal H}\ominus {\mathfrak D})=0, \
{\rm dim}({\mathcal H}_{+1}\ominus {\mathfrak D})=1, \
{\rm dim}({\mathcal H}\ominus {\mathfrak D}_*)=0, \
{\rm dim}({\mathcal H}_{+1}\ominus {\mathfrak D}_*)=1, \]

where subsets

\[{\mathfrak D}=\{f\in{\mathfrak D}(A)\cap{\mathfrak D}(\tilde A) \ \vert \ Af=\tilde Af\}, \
{\mathfrak D}_*=\{f\in{\mathfrak D}(A)\cap{\mathfrak D}(\tilde A^*) \ \vert \ Af=\tilde A^*f\}
\] 

dense both in \[{\mathcal H}.\]

Methods. Known T. Kato theorem is used to the operator \[\tilde{A}.\]

Results. Using the explicit description of \[\tilde{A}\] we prove the existence of wave operators with \[\vert\alpha\vert<\infty,\]  corresponding \[\tilde{A}.\] Wave operators are defined by equality

\[
(\tilde W_{\pm}u,v)=(u,v)\mp\frac{\alpha}{2\pi i}
\int \limits_{-\infty}^{+\infty}
\langle R_{\lambda\pm i0}u,\omega_1\rangle\langle\omega_2,\tilde R_{\lambda\mp i0}^*v\rangle
d\lambda, \quad u,v\in{\mathcal H},
\] 

where \[\tilde{R}\] and \[R\] are resolvents of perturbed and unperturbed operators.

Conclusions. The existence of wave operators is provided and their form of action is given for singular perturbation of rank one nonsymmetric potential.

Keywords


Singular perturbation; Eigenvalue problem; Krein’s formula; Nonselfadjoint perturbation; Wave operators

References


S. Albeverio et al., Solvable Models in Quantum Mechanics, 2nd ed., appendix by P. Exner. Providence, RI: AMS Chelsea Publishing, 2005, 488 p.

S. Albeverio and P. Kurasov, “Singular perturbations of differential operators. solvable schrödinger type operators”, in London Math. Soc. Lecture Note, Ser. 271. Cambridge, GB: Cambridge University Press, 2000.

V. Koshmanenko, “Singular quadratic forms in perturbation theory”, in Mathematics and its Applications, vol. 474, 1999.

Y. Berezansky and J. Brasche, “Generalized selfadjoint operators and their singular perturbations”, Methods Funct. Anal. Topology, vol. 7, no. 3, pp. 54–66, 2001.

S. Albeverio et al., “Inverse spectral problems for nonlocal Sturm-Liouville operators”, Inverse Problems, vol. 23, pp. 523–535, 2007.

L. Nizhnik, “Inverse nonlocal Sturm-Liouville problem”, Inverse Problems, vol. 26, pp. 523–535, 2010.

L. Nizhnik, “Inverse spectral nonlocal problem for the first order ordinary differential equation”, Tamkang J. Math., vol. 42, no. 3, pp. 385–394, 2011.

T.I. Vdovenko and M.E. Didkin, “Strong singular perturbations of rank one by nonsymmetric potentials”, Naukovi Visti NTUU KPI, no. 4, pp. 13–17, 2014 (in Ukrainian).

N. Dunford and J.T. Schwartz, Linear Operators. Spectral Operators.Moscow,USSR: Mir, 1974, 665 p.

T. Kato, “Wave operators and similarity for some non-selfadjoint operators”, Math. Annalen, vol. 162, pp. 258–279, 1966.

T.V. Karataeva and V.D. Koshmanenko, “Generalized sum of operators”, Math. Notes, vol. 66, no 5-6, pp. 556–564, 2000.

M.E. Dudkin and V.D. Koshmanenko, “The point spectrum of self-adjoint operators that appears under singular perturbations of finite rank”, Ukr. Math. J., vol. 55, no. 9, pp. 1532–1541, 2003.


GOST Style Citations


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DOI: https://doi.org/10.20535/1810-0546.2015.4.50461

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