Perturbations of eigenvalues embedded at threshold I. One- and three-dimensional solvable models

We examine perturbations of eigenvalues and resonances for a class of multi-channel quantum mechanical model Hamiltonians describing a particle interacting with a localized spin in dimension d = 1, 3. We consider unperturbed Hamiltonians showing eigenvalues and resonances at the threshold of the continuous spectrum and we analyze the effect of various types of perturbations on the spectral singularities. We provide algorithms to obtain convergent series expansions for the coordinates of the singularities.