Wave Operators of a Rank One Singular Perturbation by Nonsymmetric Potential

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.