Hamiltonian and Linear-Space Structure for Damped Oscillators: II. Critical Points

TL;DR

This paper extends the eigenvector expansion for damped oscillators to critical points where eigenvectors merge, revealing how perturbations affect eigenvalues and ensuring small denominators cancel near criticality.

Contribution

It introduces a representation of the bilinear map at critical points with Jordan-block structure and analyzes eigenvalue shifts under perturbations.

Findings

Eigenvalues split in equiangular directions in the complex plane.

Perturbations cause eigenvalue shifts proportional to epsilon^{1/M}.

Small denominators cancel near criticality.

Abstract

The eigenvector expansion developed in the preceding paper for a system of damped linear oscillators is extended to critical points, where eigenvectors merge and the time-evolution operator $H$ assumes a Jordan-block structure. The representation of the bilinear map is obtained in this basis. Perturbations $\epsilon\Delta H$ around an $M$-th order critical point generically lead to eigenvalue shifts $\sim\epsilon^{1/M}$ dependent on only_one_ matrix element, with the $M$ eigenvalues splitting in equiangular directions in the complex plane. Small denominators near criticality are shown to cancel.