Lee model and its resolvent analysis

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.