The Lee model was originally introduced in 1954 by Tsung-Dao Lee as a toy model in Quantum Field Theory to get insight in the renormalization procedure without the use of perturbation methods. The model describes a system consisting of a fixed source, with two internal states, coupled to a bosonic field. We consider the non-relativistic version of the model, with the bosonic field consisting of three-dimensional massive bosons. Like the original one, the model suffers from an ultraviolet divergence, but it is exactly renormalizable; building on this fact, one can construct the associated renormalized Hamiltonian, defined intrinsically through an abstract Kreı̆n-type resolvent formula, and show that it is the norm-resolvent limit of regular Hamiltonians with an ultraviolet cut-off. The model enjoys conservation of the total excitation number. Hence the spectral analysis can be performed separately in each sector of fixed excitation number. We prove the validity of a HVZ-type formula for the essential spectrum in the one- and two-excitation sectors; in higher sectors we prove an upper bound for the threshold of the essential spectrum. We determine bound states below the essential spectrum threshold in the one-excitation sector, and give explicit sufficient conditions for their existence in the two-excitation sector; more generally, in every sector we obtain criteria for weak-coupling exclusion of bound states below threshold. The point spectrum below threshold in every sector with two or more excitations is described by a bounded self-adjoint nonnegative Birman--Schwinger operator.
This is a joint work with Federica Muscolino, Diego Noja, and Andrea Posilicano.