Wave Operators of a Rank One Singular Perturbation by Nonsymmetric Potential

Тетяна Іванівна Вдовенко


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.


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


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


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