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, \hat{H}_0 and \hat{H}_\varepsilon , giving rise respectively to the unperturbed and to the perturbed evolution. The Hamiltonian \hat{H}_0 does not couple the channels and has an eigenvalue embedded in the continuous spectrum. The Hamiltonian \hat{H}_\varepsilon is a small perturbation, in resolvent sense, of \hat{H}_0 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 \hat{H}_0 shifts into a resonance for \hat{H}_\varepsilon . In dimension three we analyze details of the time behavior of the projection onto the region of the spectrum close to the resonance.