Time delay for one-dimensional quantum systems with steplike potentials

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 Schrödinger 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 $V_{\ell }\ne V_r$, 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.

48 pages, 1 figure

References in corpus (1)

• Charge Fluctuations in Quantum Point Contacts and Chaotic Cavities in the Presence of Transport

Cited by in corpus (18)

• Moments of the Wigner delay times
• The effect of time-dependent coupling on non-equilibrium steady states
• Analysis of averaged multichannel delay times
• Generalized definition of time delay in scattering theory
• The statistics of Wigner delay time in Anderson disordered systems
• Anisotropic Lavine's formula and symmetrised time delay in scattering theory
• Rectangular Potentials in a Semi-Harmonic Background: Spectrum, Resonances and Dwell Time
• Time delay for dispersive systems in quantum scattering theory
• Time in dissipative tunneling: subtleties and applications
• Extreme Covariant Quantum Observables in the Case of an Abelian Symmetry Group and a Transitive Value Space
• On a new formula relating localisation operators to time operators
• Control of Arrival Time using Structured Wave Packets
• Quantum time delay for unitary operators: general theory
• Time delay is a common feature of quantum scattering theory
• Cross section versus time delay and trapping probability
• Spectral analysis and time-dependent scattering theory on manifolds with asymptotically cylindrical ends
• Time delay for an abstract quantum scattering process
• Time delay for dispersive systems in quantum scattering theory. I. The Friedrichs model