Unfolding a degeneracy point of two unbound states: crossings and anticrossings of energies and widths

Abstract

We show that when an isolated doublet of un- bound states of a physical system becomes degenerate, the eigenenergy surfaces have an algebraic branch point of rank one and branch cuts in its real and imaginary parts starting at the same exceptional point but extending in opposite directions in parameter space. Associated with this singularity in parameter space, the scattering matrix, , and the Green's function, , have one double pole in the unphysical sheet of the complex energy plane. We characterize the universal unfolding or deformation of a typical degeneracy point of two unbound states in parameter space by means of a universal 2-parameter family of functions which is contact equivalent to the pole position function of the isolated doublet of resonances at the exceptional point and includes all small perturbations of the degeneracy condition up to contact equivalence. The rich phenomenology of crossings and anticrossings of energies and widths, as well as the sudden change in shape of the matrix pole trajectories, observed in an isolated doublet of resonances when one control parameter is varied, is fully explained in terms of the topological properties of the energy hypersurfaces close to the degeneracy point. I. INTRODUCTION In this paper, we will be concerned with some physical and mathematical aspects of the mixing and degeneracy of two unbound energy eigenstates in an isolated doublet of resonances of a quantum system depending on two control parameters.