Quantum energy levels and classical periodic orbits: discreteness and statistical duality

It is shown that both universal and non-universal correlations must exist between classical Periodic orbits in order that Gutzwillers semiclassical trace formula is consistent With a real, discrete quantum energy spectrum. Formulae for the two-point correlations are derived. The universal correlations are consistent with those conjectured by Argaman et al. (1993). Likewise, both universal and non-universal correlations must exist between quantum energy levels in order that the trace formula be consistent with the fact that periodic orbit actions are real and discrete. In this case, the two-point correlations implied are consistent with random matrix theory and previous semiclassical calculations. These ideas are illustrated with reference to the primes and the Riemann zeros.