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

A family H mu(p), mu > 0, p T-is an element of(2) of the Friedrichsmodelswith the perturbation of rank one, associated to a system of two particles, moving on the two-dimensional lattice 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 is an element of U-delta(0) subset of T-2 is proved. The analyticity of corresponding eigenfunction is shown.