On Friedrichs Model with Two Continuum States

TL;DR

This paper analyzes the Friedrichs model with multiple continua, providing explicit solutions for eigenstates and revealing complex pole structures, including virtual and resonance states, affecting the model's completeness.

Contribution

It offers an exact solution for the Friedrichs model with multiple continua and explores the resulting complex pole structures and their implications.

Findings

Discrete state generates multiple virtual and resonance poles.

Form factors can create new states on different Riemann sheets.

All generated states contribute to the generalized completeness relation.

Abstract

The Friedrichs model with one discrete state coupled to more than one continuum is studied. The exact eigenstates for the full Hamiltonian can be solved explicitly. The discrete state is found to generate more than one virtual state pole or more than one pair of resonance poles in different Riemann sheets in different situations. The form factors could also generate new states on different sheets. All these states can appear in the generalized completeness relation.