Journal of Mathematical Physics

Time delay and dilation properties in scattering theory

Heide Narnhofer

Published Apr 1, 1984• 14 citations

Abstract

We show the identity of the Eisenbud–Wigner time delay and that of Lavine for potentials V=V1+V2, V1∼1/‖x‖1+ε, x∇V1∼1/‖x‖1+ε, V2∼1/‖x‖2+ε for x→∞. Finally we show the additivity of time delay for the same class of potentials.

DOI: 10.1063/1.526216

Related papers

Time delay for dispersive systems in quantum scattering theory. I. The Friedrichs model

Rafael Tiedra de Aldecoa

We present a method for proving the existence of time delay (defined in terms of sojourn times) as well as its identity with Eisenbud-Wigner time delay in the case of the Friedrichs model. We show that this method applies to scattering by finite rank potentials.

arXiv:0804.1349Apr 8, 20080 citations

Time delay is a common feature of quantum scattering theory

Serge Richard, Rafael Tiedra de Aldecoa

We prove that the existence of time delay defined in terms of sojourn times, as well as its identity with Eisenbud-Wigner time delay, is a common feature of two Hilbert space quantum scattering theory. All statements are model-independent.

arXiv:1008.3433Aug 20, 20101 citation

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

Rafael Tiedra de Aldecoa

Nov 6, 20070 citations

Time delay for one-dimensional quantum systems with steplike potentials

Werner O. Amrein, Ph. Jacquet

arXiv:quant-ph/0610198Physical Review AFeb 21, 200727 citations

Time delay for an abstract quantum scattering process (Spectral and Scattering Theory and Related Topics)

Serge Richard

In this short review paper, we discuss the concept of time delay for an abstract quantum scattering system.Its definition in terms of sojoum times is explained as well as its identity with the so-called Eisenbud-Wigner time delay.Necessary and natural conditions for such a constmction are introduced and thoroughly discussed.Assumptions and statements are pre- cisely formulated but proofs are contained in two companion papers written in collaboration with R. Tiedra de Aldecoa.

数理解析研究所講究録Sep 1, 20110 citations

Configuration space properties of the scattering operator and time delay for potentials decaying like $\vert \chi \vert^{-\alpha}, \, \alpha > 1$

Abhisek Mohapatra, Kalyan B. Sinha, Werner O. Amrein

Annales De L Institut Henri Poincare-physique TheoriqueJan 1, 19929 citations

Journal of Mathematical Physics

Time delay and dilation properties in scattering theory

Heide Narnhofer

Published Apr 1, 1984• 14 citations

Publisher

Abstract

DOI: 10.1063/1.526216

Related papers

Time delay for dispersive systems in quantum scattering theory. I. The Friedrichs model

Rafael Tiedra de Aldecoa

arXiv:0804.1349Apr 8, 20080 citations

Time delay is a common feature of quantum scattering theory

Serge Richard, Rafael Tiedra de Aldecoa

arXiv:1008.3433Aug 20, 20101 citation

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

Rafael Tiedra de Aldecoa

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.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.

Nov 6, 20070 citations

Time delay for one-dimensional quantum systems with steplike potentials

Werner O. Amrein, Ph. Jacquet

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 $V(x)$ converging to different limits ${V}_{\ensuremath{\ell}}$ and ${V}_{r}$ 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 $V(x)$ 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.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 $V(x)$ converging to different limits ${V}_{\ensuremath{\ell}}$ and ${V}_{r}$ 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 $V(x)$ 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.

arXiv:quant-ph/0610198Physical Review AFeb 21, 200727 citations

Time delay for an abstract quantum scattering process (Spectral and Scattering Theory and Related Topics)

Serge Richard

数理解析研究所講究録Sep 1, 20110 citations

Configuration space properties of the scattering operator and time delay for potentials decaying like $\vert \chi \vert^{-\alpha}, \, \alpha > 1$ Configuration space properties of the scattering operator and time delay for potentials decaying like $\vert \chi \vert^{-\alpha}, \, \alpha > 1$

Abhisek Mohapatra, Kalyan B. Sinha, Werner O. Amrein

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 relationThe 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

Annales De L Institut Henri Poincare-physique TheoriqueJan 1, 19929 citations

|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.

V(x)

converging to different limits ${V}_{\ensuremath{\ell}}$ and

{V}_{r}

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)

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

\vert \chi \vert^{-\alpha}, \, \alpha > 1