Singular rank one non-symmetric perturbations of self-adjoint operator

Abstract

We present a construction and discuss the eigenvalue problem for a rank one singular nonselfadjoint perturbation $\tilde A = A + \alpha \langle \cdot, \omega_1 \rangle \omega_2$ of a selfadjoint half-bounded operator $A$, i.e. the operator $A$ perturbed by nonsymmetric potential $(\omega_1 \ne \omega_2)$. We give the constructive description of the operator $\tilde A$ and investigate the point spectrum that possess the operator $\tilde A$ in case of weakly singular perturbations.