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

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.