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

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 $\mathcal{PT}$.