XXVI Workshop on Geometric Methods in Physics 1-7.07.2007

Alfonso Mondragon

Geometric Aspects of the Non-Hermitian Degeneracy of Quantum Resonances

The rich phenomenology observed in the coherent mixing and degeneracy of an isolated doublet of quantum resonances (unbound energy eigenstates) when the control parameters of the system are varied is explained as due to the singularities of the energy surfaces in parameter space. More precisely, it will be shown that all the essential singularity phenomena, such as crossings and anticrossing of energies and widths, the changes of identity of the poles of the S(E)-matrix as well as the geometric or Berry phases acquired by the wave functions of the resonance states are fully explained in terms of the local topological structure of the surfaces that represent the complex energy eigenvalues in parameter space in the vicinity of a degeneracy point. As illustration, I will also show the results of a numerically exact solution of the implicit transcendental equation that defines the eigenenergy surfaces of a degenerate, isolated doublet of resonant states in the space of the real control parameters of the system in two simple but illuminating example: a) the scattering of a beam of particles by a double barrier potential with two regions of trapping; b) a two-coupled channel model of resonance reactions with square well potentials.