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 $x o -infty$ and $x o +infty$ 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 $|x| o infty$ ($V_ell = V_r$). We show that it remains true in the anisotropic case $V_ell ot = V_r$, 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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机译:本文涉及时间相关的散射理论,尤其是一类一维各向异性量子系统的时延概念。这些系统由Schr“ {o} dinger哈密顿量$ H = -Delta + V $描述,其潜在的$ V(x)$收敛到不同的限制$ V_ {ell} $和$ V_ {r} $为$ xo -infty $和$ xo + infty $。由于各向异性,它们表现出两通道结构,我们首先使用现代Mourre方法确定通道波和散射算子的存在和性质,然后使用散射理论显示了两个明显不同的时间延迟表示形式的身份,第一个以逗留时间定义,而第二个则由Eisenbud-Wigner运算符给出,这些表示形式的身份对于$ V(x )$消失为$ | x | o infty $($ V_ell = V_r $)。我们证明在各向异性情况下$ V_ell ot = V_r $仍然成立,即证明了时滞表示的时间依赖性以及它与时间独立的Eisenbud-Wigner表示的相等性。对Eisenbud-Wigner表达式进行ime依赖的解释,这在文献中通常用于时间延迟。
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