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arXiv quant-ph/0610198quant-phmath-phmath.MP
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具有阶跃势的一维量子系统的时滞

Time delay for one-dimensional quantum systems with steplike potentials

W. O. Amrein, Philippe A. Jacquet

备注：
48 pages, 1 figure
发表信息：
Phys.Rev.A75:022106,2007
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摘要：

本文研究了一维各向异性量子系统的含时散射理论，特别是时滞的概念。这些系统被描述为薛定谔哈密顿$H = -Δ+ V$与潜在的$V(x)$收敛到不同的限制$V_{\ell}$和$V_{r}$分别为$x \to -\infty$和$x \to +\infty$。由于各向异性，它们表现出双通道结构。首先，我们建立的存在性和性质的通道波和散射算子，通过使用现代Brackre方法。然后，我们使用散射理论来显示两个明显不同的表示的时间延迟的身份。第一个定义的逗留时间，而第二个是由Eisenbud-Wigner运营商。对于$V(x)$作为$|x| \to \infty$（$V_\ell = V_r$）消失的系统，这些表示的同一性是众所周知的。我们表明，它仍然是正确的各向异性的情况下，$V_\ell \not = V_r$，即我们证明了存在的时间依赖表示的时间延迟和它的平等与时间无关的Eisenbud-Wigner表示。最后，我们使用这个身份给一个依赖于时间的Eisenbud-Wigner表达式的解释，这是通常用于在文献中的时间延迟。

英文摘要：

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 Schrödinger Hamiltonian $H = -Δ+ V$ 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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