# Relativistic Lee Model and its Resolvent Analysis

## TL;DR

This paper investigates the relativistic 2+1 dimensional Lee model using resolvent analysis, establishing the self-adjointness of the Hamiltonian and deriving bounds for the ground state energy in a non-perturbative framework.

## Contribution

It provides a non-perturbative construction of the relativistic Lee model in 2+1 dimensions, proving the self-adjointness of the Hamiltonian via resolvent methods and analyzing the bound state spectrum.

## Key findings

- The resolvent defines a self-adjoint Hamiltonian.
- Bounds for the ground state energy are obtained.
- The principal operator is shown to be a self-adjoint holomorphic family.

## Abstract

We reexamine the relativistic 2+1 dimensional Lee model in light-front coordinates on flat space 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 wavefunction. The model is constructed non-perturbatively based on the resolvent formulation [1]. 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.