Return to equilibrium in theXY model

We prove that the locally perturbedXY model returns to equilibrium under the unperturbed evolution but the unperturbed model does not necessarily approach equilibrium under the perturbed evolution. In fact this latter property is false for perturbation by a local magnetization. The failure is directly attributable to the formation of bound states. If the perturbation is quadratic these problems are reduced to spectral analysis of the one-particle Hamiltonian. We demonstrate that the perturbed Hamiltonian has a finite set of eigenvalues of finite multiplicity together with some absolutely continuous spectrum. Eigenvalues can occur in the continuum if, and only if, the perturbation dislocates the system. Singular continuous spectrum cannot occur.