Warming up: Oscillators

Abstract

In order to become familiar with the language of trace formulae and in particular with the role spin plays in this context we will now describe some simple systems, namely the harmonic oscillator and generalisations, for which there exist exact trace formulae. These will be obtained in a direct way by applying the Poisson summation formula to the exact quantum mechanical density of states and only then be interpreted in terms of classical objects. The general derivation of semiclassical trace formulae certainly works the other way round. In a second step we will also demonstrate how semiclassical trace formulae can be applied in order to gain information on spectral statistics. In particular we introduce the two-point correlation function and its Fourier transform -- the spectral form factor -- and explain the idea of the diagonal approximation, which in these simple cases is exact.