 08179 Rafael Tiedra de Aldecoa
 Time delay for dispersive systems in quantum scattering theory
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Oct 6, 08

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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 hypoelliptictype. For instance $h(P)$ can be one of the usual elliptic operators such as the Schr\"odinger operator $h(P)=P^2$ or the squareroot KleinGordon 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 EisenbudWigner time delay plus a contribution due to the nonradial 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 onedimensional 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.
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