Existence of ground state eigenvalues for the spin-boson model with critical infrared divergence and multiscale analysis

A two-level atom coupled to the radiation field is studied. First principles in physics suggest that the coupling function, representing the interaction between the atom and the radiation field, behaves like |k|−1/2, as the photon momentum k tends to zero. Previous results on non-existence of ground state eigenvalues suggest that in the most general case binding does not occur in the spin-boson model, i.e., the minimal energy of the atom-photon system is not an eigenvalue of the energy operator. Hasler and Herbst have shown [12], however, that under the additional hypothesis that the coupling function be off-diagonal - which is customary to assume - binding does indeed occur. In this paper an alternative proof of binding in case of off-diagonal coupling is given, i.e., it is proven that, if the coupling function is off-diagonal, the ground state energy of the spin-boson model is an eigenvalue of the Hamiltonian. We develop a multiscale method that can be applied in the situation we study, with the help of a key symmetry operator which we use to demonstrate that the most singular terms appearing in the multiscale analysis vanish.