Fluctuations of doublet splittings using the annular billiard

The statistical distribution of quantum energy splittings due to a dynamical tunneling was investigated. It was observed that the annular billiard had whispering quasimodes, due to a discrete symmetry that existed even when chaos was presented in the underlying classical dynamics. It was also observed that the symmetric and antisymmetric combinations of the quasimodes correspond to quantum doublet states whose degeneracies decreased as the circles become more eccentric. It was found that the numerical ensembles were made, which were composed of splittings for two distinct regimes such as a semiclassical for high quantum numbers and a quantal for low quantum numbers and low energies.