Lee model and its resolvent analysis

Abstract

We revisit the relativistic (2+1)-dimensional Lee model on flat space in light-front coordinates and on a space-time with a spatial section given by a compact manifold, in the usual canonical formalism. The simpler 2+1 dimension is chosen because renormalization is needed only for the mass difference but not required for the coupling constant and the wave function. The model is constructed non-perturbatively based on the resolvent formulation [B. T. Kaynak and O. T. Turgut, The relativistic Lee model on Riemannian manifolds, J. Phys. A: Math. Theor. 42(22) (2009) 225402]. The bound state spectrum is studied through its “principal operator” and bounds for the ground state energy are obtained. We show that the formal expression found indeed defines the resolvent of a self-adjoint operator–the Hamiltonian of the interacting system. Moreover, we prove an essential result that the principal operator corresponds to a self-adjoint holomorphic family of type-A, in the sense of Kato.