  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.
