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This paper concerns time-dependent scattering theory and in particular theconcept of time delay for a class of one-dimensional anisotropic quantumsystems. 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 theanisotropy they exhibit a two-channel structure. We first establish theexistence and properties of the channel wave and scattering operators by usingthe modern Mourre approach. We then use scattering theory to show the identityof two apparently different representations of time delay. The first one isdefined in terms of sojourn times while the second one is given by theEisenbud-Wigner operator. The identity of these representations is well knownfor systems where $V(x)$ vanishes as $|x| \to \infty$ ($V_\ell = V_r$). We showthat it remains true in the anisotropic case $V_\ell \not = V_r$, i.e. we provethe existence of the time-dependent representation of time delay and itsequality with the time-independent Eisenbud-Wigner representation. Finally weuse this identity to give a time-dependent interpretation of theEisenbud-Wigner expression which is commonly used for time delay in theliterature.Comment: 48 pages, 1 figur

Time delay for one-dimensional quantum systems with steplike potentials