Time delay for dispersive systems in quantum scattering theory
Abstract
We consider time delay and symmetrised time delay (defined in terms of sojourn times) for quantum scattering pairs \H0=h(P),H\, where h(P) a dispersive operator of hypoelliptic-type. For instance h(P) can be one of the usual elliptic operators such as the Schr\"odinger operator h(P)=P2 or the square-root Klein-Gordon operator h(P)=1+P2. We show under general conditions that the symmetrised time delay exists for all smooth even localization functions. It is equal to the Eisenbud-Wigner time delay plus a contribution due to the non-radial component of the localization function. If the scattering operator S commutes with some function of the velocity operator ∇ h(P), then the time delay also exists and is equal to the symmetrised time delay. As an illustration of our results we consider the case of a one-dimensionnal Friedrichs Hamiltonian perturbed by a finite rank potential. Our study put into evidence an integral formula relating the operator of differentiation with respect to the kinetic energy h(P) to the time evolution of localization operators.