Perturbations of eigenvalues embedded at threshold: Two-dimensional solvable models

We consider a family of Hamiltonians describing a quantum particle interacting, via zero-range forces, with a localized spin in the plane. A sub-class of Hamiltonians shows eigenvalues and/or resonances at the threshold of the continuous spectrum and generates particle dynamics not affecting the localized spin. We examine the effect of a small particle-spin interaction on the spectral structure of such unperturbed Hamiltonians.