数理解析研究所講究録

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

Serge Richard

Published Sep 1, 2011• 0 citations

Abstract

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.

Related papers

Time delay for an abstract quantum scattering process

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 sojourn times is explained as well as its identity with the so-called Eisenbud-Wigner time delay. Necessary and natural conditions for such a construction are introduced and thoroughly discussed. Assumptions and statements are precisely formulated but proofs are contained in two companion papers written in collaboration with R. Tiedra de Aldecoa.

arXiv:1103.3901Mar 20, 20110 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

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 of classical and quantum scattering processes: a conceptual overview and a general definition

Massimiliano Sassoli de Bianchi

arXiv:1010.5329Open PhysicsDec 12, 201112 citations

Time-delay of classical and quantum scattering processes: a conceptual overview and a general definition

Massimiliano Sassoli de Bianchi

Jan 1, 201212 citations

Time delay for one-dimensional quantum systems with steplike potentials

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

数理解析研究所講究録

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

Serge Richard

Published Sep 1, 2011• 0 citations

OpenAlex

Abstract

Related papers

Time delay for an abstract quantum scattering process

Serge Richard

arXiv:1103.3901Mar 20, 20110 citations

Time delay is a common feature of quantum scattering theory

Serge Richard, Rafael Tiedra de Aldecoa

arXiv:1008.3433Aug 20, 20101 citation

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 of classical and quantum scattering processes: a conceptual overview and a general definition

Massimiliano Sassoli de Bianchi

Abstract We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of “fuzzy” and “free-flight” sojourn times that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they should not be called “times.”Abstract We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of “fuzzy” and “free-flight” sojourn times that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they should not be called “times.”

arXiv:1010.5329Open PhysicsDec 12, 201112 citations

Time-delay of classical and quantum scattering processes: a conceptual overview and a general definition

Massimiliano Sassoli de Bianchi

We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of "fuzzy" and "free-flight" sojourn times, that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they shouldn't be called "times".We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of "fuzzy" and "free-flight" sojourn times, that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they shouldn't be called "times".

Jan 1, 201212 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

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.