From b1cd3ef2de3f7c42facbb01b091fe93ce1377c34 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 28 Nov 2020 21:53:41 +0100 Subject: [PATCH] added small paragraph on singularities --- Manuscript/EPAWTFT.tex | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index b5518aa..8e6b70d 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -191,6 +191,13 @@ More importantly here, although EPs usually lie off the real axis, these singula Through a complex-scaling of the electronic or atomic coordinates,\cite{Moiseyev_1998} or by introducing a complex absorbing potential,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonance states are transformed into square-integrable wave functions that allow the energy and lifetime of the resonance to be computed. We refer the interested reader to the excellent book of Moiseyev for a general overview. \cite{MoiseyevBook}} +\titou{Discussion around the different types of singularities in complex analysis. +At a singular point, a function and/or its derivatives becomes infinite or undefined (hence non analytic). +One very common type of singularities (belonging to the family of isolated singularities) are poles where the function behaves $1/(\lambda - \lambda_c)^n$ where $n \in \mathbb{N}^*$ is the order of the pole. +Another class of singularities are branch points resulting from a multi-valued function such as a square root or a logarithm function and usually implying the presence of so-called branch cuts where the function ``jumps'' from one value to another. +Yet another family of singularities are formed by critical points which lie on the real axis, have more complicated functional forms and where the nature of the function undergoes a sudden transition. +} + %%%%%%%%%%%%%%%%%%%%%%% \section{Exceptional Points in Electronic Structure} \label{sec:EPs}