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\"odinger Hamiltonian $H=\ensuremath{-}\ensuremath{\Delta}+V$ with a potential
V(x) converging to different limits ${V}_{\ensuremath{\ell}}$ and
Vr as $x\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ and $+\ensuremath{\infty}$, 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 $\ensuremath{\mid}x\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ $({V}_{\ensuremath{\ell}}={V}_{r})$. We show that it remains true in the anisotropic case ${V}_{\ensuremath{\ell}}\ensuremath{\ne}{V}_{r}$, 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.
Abstract Motivated by recent experimental advances on dipolar atoms, we study the Bose gas with anisotropic short-range interaction theoretically. Starting with an anisotropic square well potential, we demonstrate that near the zero-crossing of the s -wave scattering length, there always exists a d -wave resonance. We find that the inter-channel scattering can be characterized by a scattering length. To explore the many-body effect of the short-range anisotropic inter-particle potential, we construct a pseudo-potential which captures the low-energy physics of the raw potential properly. By means of the pseudo-potential, we find the resonant d -wave interaction can inhibit the collapse of the Bose gas induced by the s -wave and dipole–dipole interaction, leading to the formation of the quantum droplet.