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 components 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 (2011) have shown, however, that a ground state does exist if the absence of self-interactions is assumed. Bach, Ballesteros, Könenberg, and Menrath (2017) have then explicitly constructed the ground state this specific case using the multiscale analysis known as Pizzo's method (Pizzo, 2003). In the present paper, a resonance eigenvalue of the complex deformed Hamiltonian is constructed by applying an inductive multiscale approach developed by Bach, Ballesteros, and Pizzo (2013, 2017), thus showing the existence of resonances in the critical case. Although the proof does not mainly rely on spectral renormalization, some techniques enter in the form of a suitable Feshbach-Schur map to control the exponential decay of the terms in the Neumann series expansions. Overall, the interaction terms obtained only involve second powers of the creation and annihilation operators and thus allow for a simpler proof.