ABSTRACT
This chapter discusses M. C. Gutzwiller's trace formula for isolated orbits. It provides its derivation in several steps going into quite some detail, because it is a central result of the periodic orbit theory, and the understanding of the main arguments and approximations used in the derivation give keys for further developments. The great formal achievement of the trace formula is that it expresses the oscillating part of the quantum level density in terms of properties of classical orbits. In a finite interval around a bifurcation point, the Gutzwiller trace formula for isolated orbits cannot be used, because it diverges at the bifurcation point. Bifurcations of periodic orbits provide the main obstacle of quantizing systems with mixed classical dynamics. The chapter shows that the inverse-squared period under the summation accelerates the convergence appreciably, and both the longer orbits and the higher repetitions of all orbits are naturally suppressed.
