Configuration properties of S-matrix and time-delay for potentials decaying like |χ| −α , α >1

Abhisek Mohapatra, Kalyan B. Sinha, W. O. Amerin

Published Jan 1, 1992• 0 citations

Abstract

The time-delay in a simple quantum mechanical scattering process is a simple 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| −α (α>1) at infinity, we use a smooth localization function with the tail growing like r 1−α (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.

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Configuration properties of S-matrix and time-delay for potentials decaying like |χ| −α , α >1

Abhisek Mohapatra, Kalyan B. Sinha, W. O. Amerin

Published Jan 1, 1992• 0 citations

OpenAlex

Abstract

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

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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

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Time delay is a common feature of quantum scattering theory

Serge Richard, Rafael Tiedra de Aldecoa

arXiv:1008.3433Aug 20, 20101 citation

Configuration space properties of the scattering operator and time delay for potentials decaying like

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

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.