# Wave Operators of a Rank One Singular Perturbation by Nonsymmetric Potential

## Authors

• Тетяна Іванівна Вдовенко NTUU KPI, Ukraine

## Keywords:

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

## 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.

## Author Biography

### Тетяна Іванівна Вдовенко, NTUU KPI

Tetiana I. Vdovenko.

PhD student

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