 06362 W. O. Amrein, Ph. Jacquet
 Time delay for onedimensional quantum systems with steplike potentials
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Dec 14, 06

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Abstract. This paper concerns timedependent scattering theory and in particular the concept of time delay for a class of
onedimensional 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 $x \rightarrow \infty$ and $x \rightarrow +\infty$ respectively. Due to the anisotropy they exhibit a twochannel 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
EisenbudWigner operator. The identity of these representations is well known for
systems where $V(x)$ vanishes as $x
\rightarrow \infty$ ($V_\ell = V_r$). We show that it remains true in the anisotropic case $V_\ell \not = V_r$, \ie we prove the existence of the
timedependent representation of time delay and its equality with
the timeindependent EisenbudWigner representation. Finally we use this identity to give a timedependent interpretation of
the EisenbudWigner expression which is commonly used for time delay in the literature.
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