# Thermal contribution of unstable states

## TL;DR

This paper investigates the thermodynamic role of unstable states within the Lee model, analyzing phase shift derivatives and spectral functions to accurately account for their contributions.

## Contribution

It provides an exact analysis of unstable state contributions to thermodynamics using the Lee model, connecting phase shifts, spectral functions, and Green functions.

## Key findings

- The phase shift derivative correctly identifies interaction contributions.
- Spectral functions and phase shifts are explicitly connected in the Lee model.
- The approach clarifies how unstable states influence thermodynamic properties.

## Abstract

Within the framework of the Lee model, we analyze in detail the difference between the energy derivative of the phase shift and the standard spectral function of the unstable state. The fact that the model is exactly solvable allows us to demonstrate the construction of these observables from various exact Green functions. The connection to a formula due to Krein, Friedal, and Lloyd is also examined. We also directly demonstrate how the derivative of the phase shift correctly identifies the relevant interaction contributions for consistently including an unstable state in describing the thermodynamics.