Existence of Resonances for the Spin-Boson-Model with Critical Coupling Function

TL;DR

This paper proves the existence of resonances in the critical spin-boson model with a specific coupling function, extending previous work on ground states to include resonance eigenvalues using multiscale analysis.

Contribution

It demonstrates the existence of resonance eigenvalues for the critical spin-boson model using multiscale analysis and complex deformation techniques.

Findings

Resonance eigenvalues are constructed for the critical coupling case.

Multiscale analysis and Feshbach-Schur map are effectively used.

The results extend understanding of spectral properties in quantum field models.

Abstract

A two-level atom coupled to the quantized radiation field is studied. In the physical relevant situation, the coupling function modeling the interaction between the two component behaves like $|k|^{-1/2}$, as the photon momentum tends to zero. This behavior is referred to as critical, as it constitutes a borderline case. Previous results on non-existence state that, in the general case, neither a ground state nor a resonance exists. Hasler and Herbst have shown [10], however, that a ground state does exist if the absence of self-interactions is assumed. Bach, Ballesteros, K\"onenberg, and Menrath have then explicitly constructed the ground state this specific case [2] using the multiscale analysis known as Pizzo's Method [13]. Building on this result, the existence of resonances is considered. In the present paper, using multiscale analysis, a resonance eigenvalue of the complex deformed Hamiltonian is constructed. Neumann series expansions are used in the analysis and a suitable Feshbach-Schur map controls the exponential decay of the terms in the series.