On nonsymmetric rank one singular perturbations of selfadjoint operators

We consider nonsymmetric rank one singular perturbations of a selfadjoint operator, i.e., an expression of the form $\tilde A = A + \alpha\left\langle\cdot, \omega_1\right\rangle\omega_2$,$\omega_1\not = \omega_2$,$\alpha\in{\mathbb C}$,inageneralcase$\omega_1,\omega_2\in{\mathcal H}_{-2}$. 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.