One-Boson Scattering Processes in the massive Spin-Boson Model

TL;DR

This paper extends the analysis of the Spin-Boson model's scattering processes to include massive bosons, overcoming spectral complications with new spectral analysis techniques to establish a scattering kernel formula.

Contribution

It introduces a novel approach using Mourre theory and Feshbach-Schur map to analyze the massive Spin-Boson model, replacing complex dilation methods.

Findings

Established a scattering kernel formula for the massive case.

Developed a spectral analysis method using Mourre theory.

Identified spectral features related to boson mass.

Abstract

The Spin-Boson model describes a two-level quantum system that interacts with a second-quantized boson scalar field. Recently the relation between the integral kernel of the scattering matrix and the resonance in this model has been established in [14] for the case of massless bosons. In the present work, we treat the massive case. On the one hand, one might rightfully expect that the massive case is easier to handle since, in contrast to the massless case, the corresponding Hamiltonian features a spectral gap. On the other hand, it turns out that the non-zero boson mass introduces a new complication as the spectrum of the complex dilated, free Hamiltonian exhibits lines of spectrum attached to every multiple of the boson rest mass energy starting from the ground and excited state energies. This leads to an absence of decay of the corresponding complex dilated resolvent close to the real line, which, in [14], was a crucial ingredient to control the time evolution in the scattering regime. With the new strategy presented here, we provide a proof of an analogous formula for the scattering kernel as compared to the massless case and use the opportunity to provide the required spectral information by a Mourre theory argument combined with a suitable application of the Feshbach-Schur map instead of complex dilation.