Semiclassical Approach for Bifurcations in a Smooth Finite-Depth Potential

The analytical trace formula for a dense cascade of bifurcations was derived using the improved stationary phase method based on extensions of the semiclassical Gutzwiller path integral approach. For the integrable version of the famous Hénon-Heiles Hamiltonian, our analytical trace formula solves all bifurcation problems, in particular, in the harmonic oscil-lator limit and the potential barrier limit. We obtain nice agreement with quantum results for gross to finer shell structures in level densities and for the shell structure energies, even near the potential barrier where there is a rather dense sequence of bifurcations. The Gutzwiller trace formula1),2) and its extensions to continuous symmetries3)–9) are nice tools to study shell structures in finite fermionic systems. This is the so-called periodic orbit theory (POT), which relates quantum fluctuations in single-particle level densities to classical periodic orbits through their dynamical charac