Virtual states and generalized completeness relation in the Friedrichs Model

TL;DR

This paper analyzes the Friedrichs model, revealing the existence of virtual-state poles below the continuum threshold and discussing their impact on the generalized completeness relations involving bound, virtual, and resonant states.

Contribution

It provides a thorough examination of pole behaviors in the Friedrichs model, highlighting the conditions for virtual states and higher-order poles, and extends the completeness relation framework.

Findings

Virtual-state poles accompany bound states below the continuum threshold.

Higher-order poles can form from merging second-sheet poles.

The generalized completeness relation includes bound, virtual, and resonant states.

Abstract

We study the well-known Friedrichs model, in which a discrete state is coupled to a continuum state. By examining the pole behaviors of the Friedrichs model in a specific form factor thoroughly, we find that, in general, when the bare discrete state is below the threshold of the continuum state, there should also be a virtual-state pole accompanying the bound-state pole originating from the bare discrete state as the coupling is turned on. There are also other second-sheet poles originating from the singularities of the form factor. We give a general argument for the existence of these two kinds of states. As the coupling is increased to a certain value, the second-sheet poles may merge and become higher-order poles. We then discuss the completeness relations incorporating bound states, virtual states, and resonant states corresponding to higher-order poles.