Analytical tests of Gutzwiller's trace formula for harmonic-oscillator potentials.

Abstract

Gutzwiller's trace formula yields the exact quantum-mechanical single-particle level density in two- and more-dimensional harmonic-oscillator potentials with irrational frequency ratios, where only isolated stable periodic orbits exist. By analytical continuation, the results are shown to hold also for oscillators with commensurable frequencies and with continuous symmetries, where most classical orbits are degenerate; the divergences of the trace formula occurring in these cases are shown to cancel exactly. In the two-dimensional case, we give a closed formula for the Maslov index that does not repeat itself in succesive periods.