Strong and weak coupling of eigenvalues of complex matrices

Abstract

The paper presents a general theory of coupling of eigenvalues of complex matrices of arbitrary dimension smoothly 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 numerical exam- ples from crystal optics illustrate effectiveness and accuracy of the presented theory. I. INTRODUCTION Behavior of eigenvalues of matrices dependent on param- eters is a problem of general interest having many important applications in natural and engineering sciences. In modern physics, e.g. quantum mechanics, crystal optics, physical chemistry, acoustics and mechanics, multiple eigenvalues in matrix spectra associated with specific effects attract great interest of researchers since the papers (1), (2). In recent papers, see e.g. (3)-(6), two important cases are distinguished: the diabolic points (DPs) and the exceptional points (EPs). From mathematical point of view DP is a point where the eigenvalues coalesce, while corresponding eigenvectors remain different; and EP is a point where both eigenvalues and eigenvectors merge forming a Jordan block. Both the DP and EP cases are interesting in applications and were observed in experiments (6), (7). In this paper we present a general theory of coupling of eigenvalues of complex matrices of arbitrary dimen- sion smoothly depending on multiple real parameters. Two essential cases of weak and strong coupling based on a Jordan form of the system matrix are distinguished. These two cases correspond to diabolic and exceptional points, respectively. We derive general formulae describing coupling and decoupling of eigenvalues, crossing and avoided crossing of eigenvalue surfaces. It is emphasized that the presented theory of coupling of eigenvalues of complex matrices gives not only qualitative, but also quantitative results on behavior of eigenvalues based only on the information taken at the singular points. The paper is based on the author's previous research on interaction of eigenvalues for matrices and differential operators depending on multiple parameters (8)- (11); for more references see the recent book (12).