Diffraction and spectral statistics in systems with a multilevel scatterer

A semiclassical framework to interpret the spectral rigidity of a system containing a scatterer with internal states is developed. Our prototype system is a scaled Rydberg molecule in an external magnetic field, where the core is a multilevel scatterer: the potential sheet in which the outer electron moves depends on the quantum state of the core. Thus the electron-core collision, interpreted in terms of the diffraction of the semiclassical waves associated with the outer electron on the core, can result in a change of the electron's dynamical regime. We examine the contribution of the diffraction to the spectral rigidity by obtaining the diffractive Green's function in the semiclassical limit. We concurrently determine this contribution from accurate quantum spectra and compare numerically the semiclassical and quantum results. Our findings indicate that, in a system with a multilevel scatterer, the diffractive contribution to the spectral rigidity cannot be accounted for by a simple universal expression, but rather depends on system specific nonuniversal terms: the quantum properties of the scatterer (reflected by the relative values of the phase shifts in the different channels) and the classical properties of the shortest periodic orbits in the different dynamical regimes.