Semiclassical trace formula for the two-dimensional radial power-law potentials

TL;DR

This paper derives a semiclassical trace formula for two-dimensional radial power-law potentials, accurately capturing quantum shell structures and bifurcation effects, with analytical results for specific powers and various limits.

Contribution

It presents an improved stationary phase method-based trace formula for radial power-law potentials, including analytical results for powers 4 and 6, and addresses bifurcation phenomena.

Findings

Accurately reproduces quantum shell structures.

Effectively describes bifurcation phenomena.

Unified analytical trace formula for various potential limits.

Abstract

The trace formula for the density of single-particle levels in the two-dimensional radial power-law potentials, which nicely approximate the radial dependence of the Woods-Saxon potential and quantum spectra in a bound region, was derived by the improved stationary phase method. The specific analytical results are obtained for the powers 4 and 6. The enhancement phenomena near the bifurcations of periodic orbits are found to be significant for the description of the fine shell structure. It is shown that the semiclassical trace formulas for the shell corrections to the level density and energy reproduce the quantum results with good accuracy through all the bifurcation (symmetry breaking) catastrophe points, where the standard stationary-phase method breaks down. Various limits (including the harmonic oscillator and the spherical billiard) are obtained from the same analytical trace formula.