On the nonrelativistic Lee model

In this work we present two rigorous results on the nonrelativistic Lee model following a method proposed by Rajeev in an unpublished article (S. G. Rajeev, hep-th 9902025). Thus this short paper should be considered as a commentary on Rajeev. In the unpublished paper of Rajeev, the renormalization of the Hamiltonian is accomplished at the level of resolvents. We first establish that the renormalized resolvent of the interacting Hamiltonian indeed defines a unique closed densely defined operator acting on the free Fock space of bosons. Next we give a justification in the mean field approximation that the ground state energy is bounded from below and the system has a good thermodynamic limit by elaborating along the original arguments of Rajeev. Our arguments in two dimensions do not yield better bounds, but this could be due to the inadequacy of the method used. In both cases though the ground state energy is not significantly altered to give a nontrivial ground state energy per particle.