Resonances in models of spin-dependent point interactions

In dimension d = 1, 2, 3 we define a family of two-channel Hamiltonians obtained as point perturbations of the generator of the free decoupled dynamics. Within the family we choose two Hamiltonians, H0 and Hε, giving rise respectively to the unperturbed and to the perturbed evolution. The Hamiltonian H0 does not couple the channels and has an eigenvalue embedded in the continuous spectrum. The Hamiltonian Hε is a small perturbation, in resolvent sense, of H0 and exhibits a small coupling between the channels. We take advantage of the complete solvability of our model to prove with simple arguments that the embedded eigenvalue of H0 shifts into a resonance for Hε. In dimension three we analyze details of the time behavior of the projection onto the region of the spectrum close to the resonance