W. O. Amrein, Ph. Jacquet
Time delay for one-dimensional quantum systems with steplike potentials
(627K, Postscript)
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 $x \rightarrow -\infty$ and $x \rightarrow +\infty$ 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 $|x|
\rightarrow \infty$ ($V_\ell = V_r$). We show that it remains true in the anisotropic case $V_\ell \not = V_r$, \ie we prove the existence of the
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.