Classical explanation of Maslov indices and amplitude factors in the Berry–Tabor trace formula

Explain, using classical periodic-orbit dynamics in extended phase space for the spherical harmonic oscillator with spin–orbit coupling, the origin and values of the irrational Maslov indices and amplitude factors that arise in the Berry–Tabor trace formula constructed from the exact quantum spectrum.

Background

From the exactly solvable spectrum of the spherical harmonic oscillator with spin–orbit coupling, the Berry–Tabor formula yields contributions associated with primitive periods T_± and T_0. However, the resulting Maslov indices appear irrational and the amplitudes exhibit scalings not straightforwardly deduced from classical degeneracies or stability, creating a mismatch between quantum expressions and classical periodic-orbit characteristics.

The paper explicitly states that these Maslov indices and amplitude factors have yet to be explained via classical dynamics, pointing to a gap in the quantum–classical correspondence that calls for a dynamical derivation or interpretation rooted in the extended phase-space treatment of spin.