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

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, Könenberg, 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.

References in corpus (5)

• Resonance Theory of Decoherence and Thermalization
• Ground States in the Spin Boson Model
• Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation
• Analyticity of Resonances and Eigenvalues and Spectral Properties of the massless Spin-Boson Model
• Relation between the Resonance and the Scattering Matrix in the massless Spin-Boson Model

Cited by in corpus (4)

• Generalized spin-boson models with non-normalizable form factors
• Self-adjointness of a class of multi-spin-boson models with ultraviolet divergences
• Renormalization of spin$\unicode{x2013}$boson interactions mediated by singular form factors
• Self-adjointness and domain of generalized spin-boson models with mild ultraviolet divergences