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

Abstract

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.