TIME DELAY FOR DISPERSIVE SYSTEMS IN QUANTUM SCATTERING THEORY

Abstract

We consider time delay and symmetrized time delay (defined in terms of sojourn times) for quantum scattering pairs {H0 = h(P), H}, where h(P) is a dispersive operator of hypoelliptic-type. For instance, h(P) can be one of the usual elliptic operators such as the Schrödinger operator h(P) = P2 or the square-root Klein–Gordon operator [Formula: see text]. We show under general conditions that the symmetrized 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 symmetrized time delay. As an illustration of our results, we consider the case of a one-dimensional Friedrichs Hamiltonian perturbed by a finite rank potential. Our study puts 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.