Periodic-Orbit Theory

Abstract

[[ RM: A review paper on cycle expansions. I quote the introduction: in section (2) ]] I will summarize Gutzwiller's theory for the spectrum of eigenenergies and extend it to diagonal matrix elements as well. The derivation of the associated zeta function is given (2.2) and the identification of suitable scaling variables discussed (2.3). In section 3 tools necessary for the organization of chaos will be discussed: symbolic dynamics (3.1), the connectivity matrix (3.3), the topological zeta function (3.4) and general transfer matrices and zeta functions (3.5). Although illustrated for the case of hard collisions in a billiard, the symbolic dynamics can be extended to `smooth collisions' in smooth potentials (3.2). In systems with discrete symmetries, zeta functions factorize into zeta functions on invariant subspaces. This symmetry factorization and the associated reduction in symbolics is discussed in section 4. The ideas developed here are illustrated for the example of a free particle reflected elastically off three disks in section 5. Methods to find periodic orbits (5.1), the convergence of the trace formula (5.2), the semiclassical computation of scattering resonances (5.3), the convergence of the cycle expansion (5.4) and methods to obtain eigenvalues of the bounded billiard (5.5) are discussed. The relevant parts of a classical periodic orbit theory are developed in section 6.1, including a discussion of escape rates and the Hannay-Ozorio de Almeida sum rule (6.2). Finally, the issue of semiclassical matrix elements is taken up again and applications to experiments are discussed.