Time delay and short-range scattering in quantum waveguides

TL;DR

This paper rigorously establishes the equivalence of modified and Eisenbud-Wigner time delays in quantum waveguides, providing new spectral and scattering results in both abstract and short-range settings.

Contribution

It proves the identity of different time delay definitions and derives spectral properties, wave operators, and an explicit S-matrix formula for quantum waveguides.

Findings

Proves the equivalence of modified and Eisenbud-Wigner time delays.

Establishes spectral properties and wave operators for the Hamiltonian.

Provides an explicit formula for the S-matrix.

Abstract

Although many physical arguments account for using a modified definition of time delay in multichannel-type scattering processes, one can hardly find rigorous results on that issue in the literature. We try to fill in this gap by showing, both in an abstract setting and in a short-range case, the identity of the modified time delay and the Eisenbud-Wigner time delay in waveguides. In the short-range case we also obtain limiting absorption principles, state spectral properties of the total Hamiltonian, prove the existence of the wave operators and show an explicit formula for the S-matrix. The proofs rely on stationary and commutator methods.