Threshold Effects for the Generalized Friedrichs Model with the Perturbation of Rank One

Abstract

A family $H_{\mu }(p)$, $\mu >0$, $p\in {𝕋}^2$ of the Friedrichs models with the perturbation of rank one, associated to a system of two particles, moving on the two-dimensional lattice ${\mathbb{Z}}^2$ is considered. The existence or absence of the unique eigenvalue of the operator $H_{\mu }(p)$ lying below threshold depending on the values of $\mu >0$ and $p\in U_{\delta }(0)\subset {𝕋}^2$ is proved. The analyticity of corresponding eigenfunction is shown.