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arXiv:math-ph/0609044 · Journal of Mathematical Physics

Generalized definition of time delay in scattering theory

Christian Gérard, Rafael Tiedra de Aldecoa

Published Dec 1, 2007• 15 citations
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Abstract

We advocate for the systematic use of a symmetrized definition of time delay in scattering theory. In two-body scattering processes, we show that the symmetrized time delay exists for arbitrary dilated spatial regions symmetric with respect to the origin. It is equal to the usual time delay plus a new contribution, which vanishes in the case of spherical spatial regions. We also prove that the symmetrized time delay is invariant under an appropriate mapping of time reversal. These results are also discussed in the context of classical scattering theory.

DOI
10.1063/1.2816255

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We advocate for the systematic use of a symmetrized definition of time delay in scattering theory. In two-body scattering processes, we show that the symmetrized time delay exists for arbitrary dilated spatial regions symmetric with respect to the origin. It is equal to the usual time delay plus a new contribution, which vanishes in the case of spherical spatial regions. We also prove that the symmetrized time delay is invariant under an appropriate mapping of time reversal. These results are also discussed in the context of classical scattering theory.
We consider, in quantum scattering theory, symmetrised time delay defined in terms of sojourn times in arbitrary spatial regions symmetric with respect to the origin. For potentials decaying more rapidly than ∣x∣−4|x|^{-4}∣x∣−4 at infinity, we show the existence of symmetrised time delay, and prove that it satisfies an anisotropic version of Lavine's formula. The importance of an anisotropic dilations-type operator is revealed in our study.
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.
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 Schrödinger operator h(P) = P 2 or the square-root Klein-Gordon operator h(P) = √ 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 ∇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. 1 Introduction and main results One can find a large literature on the identity of Eisenbud-Wigner time delay and time delay in quantum scattering defined in terms of sojourn times (see [3, 7, 8, 12, 19, 23,