In the preceding chapter we derived semiclassical expressions for the quantum mechanical propagator, the Green function, and the density of states. Now we are going to discuss a number of consequences and applications, in particular of the Gutzwiller trace formula. Anybody working in the field should be familiar with the basic ideas applied in the derivation, but an understanding of the implications of periodic orbit theory is also possible for readers not willing to enter too deeply into the details. For this reason knowledge of the preceding chapter should be dispensable for the major part of the present one.

We start with a description of the techniques applied to extract the contributions of the different periodic orbits from the spectra and wave functions. The spectra of billiards and hydrogen atoms in strong magnetic fields suggest a semiclassical interpretation. The most spectacular manifestation of periodic orbits is the scarring phenomenon, found in many wave functions of chaotic billiards.

As the quantum mechanical spectrum is uniquely determined by the periodic orbits, i.e. by individual system properties, the success of random matrix theory in description of numerous spectral correlations requires an explanation. This is given by Berry in his semiclassical theory of spectral rigidity [Ber85].

It is comparatively easy to analyse a given spectrum in terms of periodic orbit theory. This is not true for the reverse procedure, namely the calculation of a spectrum from the periodic orbits of the system.