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

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.

Non-Hermitian quantum annealing in the ferromagnetic Ising model

“…The point of degeneracy is known as the exceptional point, and it is characterized by a coalescence of eigenvalues and their corresponding eigenvectors, as well. Therefore, studying the behavior of the system in the vicinity of the exceptional point requires a special care [19][20][21]. In the vicinity of the level crossing point, only the twodimensional Jordan block, related to the level crossing, makes the most considerable contribution to the quantum evolution.…”

How to Play a Disc Brake: A Dissipation-Induced Squeal

“…Consider a perturbation of the gyroscopic system , assuming that the size of the H is described by the asymptotic formula [32] Although the veering phenomenon in the systems with gyroscopic coupling was studied both numerically and analytically, e.g in [3,4,7,8,13,17,31], the explicit expressions (10) - (14) for the splitting of the double eigenvalues due to action of forces of all types were not previously derived. The approach used in our paper is distinct of that of the cited works.…”

“…The approach used in our paper is distinct of that of the cited works. It is based on the perturbation theory of multiple eigenvalues [28,32,33]. The spectrum of the perturbed system is described by means of only the derivatives of the operator with respect to parameters and the eigenvectors of the multiple eigenvalue calculated directly at a node of the spectral mesh.…”

“…J , where the rotational speed of the string is J [13]. Introducing non-dimensional variables and parameters (32) , , , , , ,…”

“…In the following using the perturbation theory of multiple eigenvalues [28,32,33,38] we will show that independently on the definiteness of the damping matrix, there exist combinations of dissipative and nonconservative positional forces, causing the flutter instability in the vicinity of the nodes of the spectral mesh for the angular velocities from the subcritical range. Zero and negative eigenvalues in the spectrum of the damping matrix encourage the development of the localized subcritical flutter instability while zero eigenvalues in the matrix of non-conservative positional forces suppress it.…”