Extensions of the Gutzwiller theory
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Extensions of the Gutzwiller theory book
The trace formula of M. C. Gutzwiller can only be applied to systems in which all classical periodic orbits are isolated. Balian and Bloch used a multi-reflection expansion of the energy-dependent Green's function to investigate first the smooth and then the oscillating of the density of eigenstates in a cavity with ideally reflecting walls. Strutinsky and coworkers generalized the Gutzwiller theory for smooth potentials with arbitrary symmetries and correspondingly degenerate periodic orbits. This chapter discusses some of these extensions of Gutzwiller's theory to systems with degenerate orbits. It presents several trace formulae for systems with continuous symmetries and illustrate them by some specific examples. Even when treatment of degenerate periodic orbits has been made possible, there still persists a generic problem with all these trace formulae. The precise number and choice of exact integrations depends on overall degeneracy of a given orbit family and particular nature and symmetry of the given problem.