Relativistic LEE model on 2+1 dimensional riemannian manifolds

Abstract

In this work, we analyze a two-level system coupled to an arbitrary number of bosons with a relativistic dispersion relation in a static background metric while ignoring pair creation processes. One can obtain a non-perturbative formulation of this problem by directly studying the associated resolvent following an idea proposed by Rajeev. The resolvent allows us to estimate the ground state energy from above and below thanks to the fact that it is formulated through an operator, so called principal operator. Whenever the eigenvalues of the principal operator hit a zero, as they flow with the energy parameter, we find a possible pole in the resolvent, which typically corresponds to a bound state in the spectrum. The rigorous study of this principal operator includes showing that this operator is a holomorphic family of type-A in the sense of Kato. This in turn justifies the fact that our resolvent formula defines a self adjoint quantum Hamiltonian as well as putting our estimates on a firmer ground. The required operator estimates are obtained through recent two sided heat kernel estimates on manifolds.