Time delay for dispersive systems in quantum scattering theory

TL;DR

This paper investigates the existence and properties of time delay in quantum scattering for dispersive systems, establishing conditions under which symmetrised time delay exists and relates to the Eisenbud-Wigner time delay, with applications to specific models.

Contribution

It extends the analysis of time delay to dispersive operators like the Schrödinger and Klein-Gordon operators, providing general conditions for existence and explicit formulas relating to localization operators.

Findings

Symmetrised time delay exists for all smooth even localization functions.

Time delay equals Eisenbud-Wigner delay plus a non-radial localization contribution.

In cases where the scattering operator commutes with a velocity function, time delay exists and matches symmetrised delay.

Abstract

We consider time delay and symmetrised time delay (defined in terms of sojourn times) for quantum scattering pairs $\{H_0=h(P),H\}$, 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 $h(P)=P^2$ or the square-root Klein-Gordon operator $h(P)=\sqrt{1+P^2}$. 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 $\nabla h(P)$, 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.