Structural invariance and the energy spectrum

We extend the application of the concept of structural invariance to bounded timeindependent systems. This concept, previously introduced by two of us to argue that the connection between random matrix theory and quantum systems with a chaotic classical counterpart is in fact largely exact in the semiclassical limit is extended to the energy spectra of bounded timeindependent systems. We proceed by showing that the results obtained previously for the quasienergies and eigenphases of the S-matrix can be extended to the eigenphases of the quantum Poincaré map which is unitary in the semiclassical limit. We then show that its eigenphases in the chaotic case move rather stiffly around the unit circle and thus their local statistical fluctuations transfer to the energy spectrum via Bogomolny’s prescription. We verify our results by studying numerically the properties of the eigenphases of the quantum Poincaré map for billiards by using the boundary integral method.