dorsal/arxiv
View SchemaTime delay for one-dimensional quantum systems with steplike potentials
| Authors | W. O. Amrein, Philippe A. Jacquet |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0610198 |
| URL | https://arxiv.org/abs/quant-ph/0610198 |
| DOI | 10.1103/PhysRevA.75.022106 |
| Journal | Phys.Rev.A75:022106,2007 |
Abstract
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\"{o}dinger Hamiltonian $H = -\Delta + V$ with a potential $V(x)$ converging to different limits $V_{\ell}$ and $V_{r}$ as $x \to -\infty$ and $x \to +\infty$ 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 $|x| \to \infty$ ($V_\ell = V_r$). We show that it remains true in the anisotropic case $V_\ell \not = V_r$, 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.
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"abstract": "This paper concerns time-dependent scattering theory and in particular the\nconcept of time delay for a class of one-dimensional anisotropic quantum\nsystems. These systems are described by a Schr\\\"{o}dinger Hamiltonian $H =\n-\\Delta + V$ with a potential $V(x)$ converging to different limits $V_{\\ell}$\nand $V_{r}$ as $x \\to -\\infty$ and $x \\to +\\infty$ respectively. Due to the\nanisotropy they exhibit a two-channel structure. We first establish the\nexistence and properties of the channel wave and scattering operators by using\nthe modern Mourre approach. We then use scattering theory to show the identity\nof two apparently different representations of time delay. The first one is\ndefined in terms of sojourn times while the second one is given by the\nEisenbud-Wigner operator. The identity of these representations is well known\nfor systems where $V(x)$ vanishes as $|x| \\to \\infty$ ($V_\\ell = V_r$). We show\nthat it remains true in the anisotropic case $V_\\ell \\not = V_r$, i.e. we prove\nthe existence of the time-dependent representation of time delay and its\nequality with the time-independent Eisenbud-Wigner representation. Finally we\nuse this identity to give a time-dependent interpretation of the\nEisenbud-Wigner expression which is commonly used for time delay in the\nliterature.",
"arxiv_id": "quant-ph/0610198",
"authors": [
"W. O. Amrein",
"Philippe A. Jacquet"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1103/PhysRevA.75.022106",
"journal_ref": "Phys.Rev.A75:022106,2007",
"title": "Time delay for one-dimensional quantum systems with steplike potentials",
"url": "https://arxiv.org/abs/quant-ph/0610198"
},
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