Existence of Ground State Eigenvalues for the SpinBoson Model with Critical Infrared Divergence and Multiscale Analysis
Abstract
A twolevel 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 $\vert k \vert^{ 1/2}$, as the photon momentum k tends to zero. Previous results on nonexistence of ground state eigenvalues suggest that in the most general case binding does not occur in the spinboson model, i.e., the minimal energy of the atomphoton 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 offdiagonal which is customary to assumebinding does indeed occur. In this paper an alternative proof of binding in case of offdiagonal coupling is given, i.e., it is proven that, if the coupling function is offdiagonal, the ground state energy of the spinboson model is an eigenvalue of the Hamiltonian. We develop a multiscale method that can be applied in the situation we study, identifying a new key symmetry operator which we use to demonstrate that the most singular terms appearing in the multiscale analysis vanish.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 DOI:
 10.48550/arXiv.1605.08348
 arXiv:
 arXiv:1605.08348
 Bibcode:
 2016arXiv160508348B
 Keywords:

 Mathematical Physics
 EPrint:
 29 pages