The Analytic Continuation of the Lippmann-Schwinger Eigenfunctions, and Antiunitary Symmetries

TL;DR

This paper reviews the proper method for analytically continuing Lippmann-Schwinger eigenfunctions into the complex plane, emphasizing the role of antiunitary symmetries in ensuring meaningful resonance theory results.

Contribution

It clarifies the correct approach to analytic continuation of Lippmann-Schwinger functions and highlights the importance of antiunitary symmetries in this process.

Findings

Naive continuation leads to nonsensical results in resonance theory.

Correct continuation relies on invariance under antiunitary symmetries.

Provides a framework for consistent resonance analysis.

Abstract

We review the way to analytically continue the Lippmann-Schwinger bras and kets into the complex plane. We will see that a naive analytic continuation leads to nonsensical results in resonance theory, and we will explain how the non-obvious but correct analytical continuation is done. We will see that the physical basis for the non-obvious but correct analytic continuation lies in the invariance of the Hamiltonian under anti-unitary symmetries such as time reversal or PT.