Rank-One Singular Perturbations with a Dual Pair of Eigenvalues

We discuss the eigen-values problem for rank one singular perturbations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tilde A = A\tilde + \alpha \langle \cdot ,\omega \rangle \omega $$ \end{document} of a self-adjoint unbounded operator A with a gap in its spectrum. We give a constructive description of operators à which possess at least two new eigenvalues, one in the resolvent set and other in the spectrum of A.