Abstract :
We review the status of the semiclassical trace formula with emphasis on the particular types of singularities that occur in the Gutzwiller-Voros zeta function for bound chaotic systems. To understand the problem better, we extend the discussion to include various classical zeta functions and we contrast properties of Axiom-A scattering systems with those of typical bound systems. Singularities in classical zeta functions contain topological and dynamical information, concerning, for example, anomalous diffusion, phase transitions among generalized Lyapunov exponents, power law decay of correlations. Singularities in semiclassical zeta functions are artifacts and enter because one neglects some quantum effects when deriving them, typically by making saddle point approximation when the saddle points are not separated enough. The discussion is exemplified by the Sinai billiard where intermittent orbits associated with neutral orbits induce a branch point in the zeta functions. This singularity is responsible for a diverging diffusion constant in Lorentz gases with unbounded horizon. In the semiclassical case there is interference between neutral orbits and intermittent orbits. The Gutzwiller-Voros zeta function exhibits a branch point because it does not take this effect into account. Another consequence is that individual states, high up in the spectrum, cannot be resolved by the Berry-Keating technique.
