Generic Cuspidal Points and Their Localization

TL;DR

This paper investigates the behavior of eigenvalues and eigenvectors of smooth complex matrix functions near cuspidal points where eigenvalues coalesce, providing methods to localize these points through phase analysis and eigenvalue periodicity.

Contribution

It introduces a rigorous approach to identify and analyze generic cuspidal points in parameter-dependent matrices, linking eigenvalue coalescence with phase accumulation and localization techniques.

Findings

Established conditions for phase accumulation around cuspidal points

Linked eigenvalue periodicity to localization of cuspidal points

Provided a theoretical framework for analyzing eigenvalue coalescence

Abstract

In this work we consider generic coalescing of eigenvalues of smooth complex valued matrix functions depending on 2 parameters. We call generic cuspidal points the parameter values where eigenvalues coalesce and we discuss the relation between cuspidal points and the closely related exceptional points studied in the literature. By considering loops in parameter space enclosing the cuspidal points, we rigorously prove when there is a phase accumulation for the eigenvectors and further detail how, by looking at the periodicity of the eigenvalues along the loop, and/or by looking at the aforementioned phase accumulation, one may be able to localize generic cuspidal points.