Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation

TL;DR

This paper develops a new theoretical framework for understanding how eigenvalue surfaces of symmetric and Hermitian matrices deform under complex perturbations near diabolic points, with applications in crystal optics.

Contribution

It introduces general asymptotic formulas for eigenvalue surface deformations caused by complex perturbations near diabolic points, expanding existing theories.

Findings

Derived asymptotic formulas for eigenvalue surface deformation

Applied theory to singularities in crystal optics

Enhanced understanding of eigenvalue behavior under complex perturbations

Abstract

The paper presents a new theory of unfolding of eigenvalue surfaces of real symmetric and Hermitian matrices due to an arbitrary complex perturbation near a diabolic point. General asymptotic formulae describing deformations of a conical surface for different kinds of perturbing matrices are derived. As a physical application, singularities of the surfaces of refractive indices in crystal optics are studied.