Coupling of eigenvalues of complex matrices at diabolic and exceptional points

TL;DR

This paper develops a comprehensive theory describing how eigenvalues of complex matrices behave near diabolic and exceptional points, including geometric interpretations and asymptotic formulas, with physical examples demonstrating its accuracy.

Contribution

It introduces a general framework for understanding eigenvalue coupling in complex matrices, distinguishing weak and strong coupling, and providing geometric and asymptotic insights.

Findings

Eigenvalue surfaces exhibit crossing and avoided crossing near special points.

The theory accurately predicts eigenvalue behavior in physical systems.

Geometric interpretations clarify eigenvalue interactions in low-dimensional spaces.

Abstract

The paper presents a general theory of coupling of eigenvalues of complex matrices of arbitrary dimension depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two physical examples illustrate effectiveness and accuracy of the presented theory.