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Anisotropic Lavine's formula and symmetrised time delay in scattering theory

Rafael Tiedra de Aldecoa

Published Nov 6, 2007• 0 citations
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Abstract

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 $|x|^{-4}$ 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.

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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 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,
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 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.
This paper concerns time-dependent scattering theory and in particular the concept of time delay for a class of one-dimensional anisotropic quantum systems. These systems are described by a Schr\"odinger Hamiltonian $H=\ensuremath{-}\ensuremath{\Delta}+V$ with a potential V(x)V(x)V(x) converging to different limits ${V}_{\ensuremath{\ell}}$ and Vr{V}_{r}Vr​ as $x\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ and $+\ensuremath{\infty}$, respectively. Due to the anisotropy, they exhibit a two-channel structure. We first establish the existence and properties of the channel wave and scattering operators by using the modern Mourre approach. We then use scattering theory to show the identity of two apparently different representations of time delay. The first one is defined in terms of sojourn times while the second one is given by the Eisenbud-Wigner operator. The identity of these representations is well known for systems where V(x)V(x)V(x) vanishes as $\ensuremath{\mid}x\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ $({V}_{\ensuremath{\ell}}={V}_{r})$. We show that it remains true in the anisotropic case ${V}_{\ensuremath{\ell}}\ensuremath{\ne}{V}_{r}$, i.e., we prove the existence of the time-dependent representation of time delay and its equality with the time-independent Eisenbud-Wigner representation. Finally, we use this identity to give a time-dependent interpretation of the Eisenbud-Wigner expression, which is commonly used for time delay in the literature.
Configuration space properties of the scattering operator and time delay for potentials decaying like ∣χ∣−α, α>1\vert \chi \vert^{-\alpha}, \, \alpha > 1∣χ∣−α,α>1
The time-delay in a simple quantum mechanical scattering process is defined to be the limit as r → ∞ of the difference of the sojourn times of a scattering state and of the associated free state in a ball of radius r, with the particle under the influence of a potential W(x). For W(x)∼|x| #75α (α>1) at infinity, we use a smooth localization function with the tail growing like r 1#75δ (δ > 0, depending on W) instead of the characteristic function of a ball of radius r and it is shown that the time-delay exists and satisfies the Eisenbud-Wigner relation