Analyticity of Resonances and Eigenvalues and Spectral Properties of the massless Spin-Boson Model

TL;DR

This paper develops a multiscale analysis method to prove the analyticity and spectral localization of resonances and eigenvalues in the massless Spin-Boson model, providing new insights into its spectral properties.

Contribution

It introduces a novel multiscale analysis approach to study analyticity and spectral localization of resonances and eigenvalues in the Spin-Boson model, a first in this context.

Findings

Resonances and eigenvalues are analytic with respect to dilation and coupling.

Spectrum near resonances is localized in two cones in the complex plane.

Norm estimates for the resolvent near resonances and eigenvalues are established.

Abstract

We extend the method of multiscale analysis for resonances introduced in [5] in order to infer analytic properties of resonances and eigenvalues (and their eigenprojections) as well as estimates for the localization of the spectrum of dilated Hamiltonians and norm-bounds for the corresponding resolvent operators, in neighborhoods of resonances and eigenvalues. We apply our method to the massless Spin-Boson model assuming a slight infrared regularization. We prove that the resonance and the ground-state eigenvalue (and their eigenprojections) are analytic with respect to the dilation parameter and the coupling constant. Moreover, we prove that the spectrum of the dilated Spin-Boson Hamiltonian in the neighborhood of the resonance and the ground-state eigenvalue is localized in two cones in the complex plane with vertices at the location of the resonance and the ground-state eigenvalue, respectively. Additionally, we provide norm-estimates for the resolvent of the dilated Spin-Boson Hamiltonian near the resonance and the ground-state eigenvalue. The topic of analyticity of eigenvalues and resonances has let to several studies and advances in the past. However, to the best of our knowledge, this is the first time that it is addressed from the perspective of multiscale analysis. Once the multiscale analysis is set up our method gives easy access to analyticity: Essentially, it amounts to proving it for isolated eigenvalues only and use that uniform limits of analytic functions are analytic. The type of spectral and resolvent estimates that we prove are needed to control the time evolution including the scattering regime. The latter will be demonstrated in a forthcoming publication. The introduced multiscale method to study spectral and resolvent estimates follows its own inductive scheme and is independent (and different) from the method we apply to construct resonances.