Geometry of mixing and degeneracy of resonances in a two-channel model

Abstract

Some geometric aspects of the mixing and degeneracy of two unbound energy eigenstates of a quantum system depending on two control parameters will be described. It will be shown that the surface that represents the complex resonance energy eigenvalues in parameter space has a singularity of square root type at the degeneracy point, and branch cuts in its real and imaginary parts that start at the exceptional point but extend in opposite directions. The rich phenomenology of crossings and anticrossings of energies and widths of the resonances in an isolated doublet of unbound states as well as the changes of identity of the poles of the S–matrix, observed when one control parameter is varied while the other is kept constant is fully explained in terms of sections of the eigenenergy surfaces.