Semiclassical quantization of the diamagnetic hydrogen atom with near action-degenerate periodic-orbit bunches

TL;DR

This paper proves the existence of periodic orbit bunches in the diamagnetic Kepler problem, showing how they can be used to improve semiclassical quantization efficiency by reducing the classical data needed for spectral calculations.

Contribution

It introduces the concept of near action-degenerate periodic orbit bunches and demonstrates their application in semiclassical quantization of the diamagnetic hydrogen atom.

Findings

Periodic orbit bunches exist for the diamagnetic Kepler problem.

Using orbit representatives reduces classical data by up to a factor of 20.

Orbit bunches enhance the efficiency of semiclassical spectral calculations.

Abstract

The existence of periodic orbit bunches is proven for the diamagnetic Kepler problem. Members of each bunch are reconnected differently at self-encounters in phase space but have nearly equal classical action and stability parameters. Orbits can be grouped already on the level of the symbolic dynamics by application of appropriate reconnection rules to the symbolic code in the ternary alphabet. The periodic orbit bunches can significantly improve the efficiency of semiclassical quantization methods for classically chaotic systems, which suffer from the exponential proliferation of orbits. For the diamagnetic hydrogen atom the use of one or few representatives of a periodic orbit bunch in Gutzwiller's trace formula allows for the computation of semiclassical spectra with a classical data set reduced by up to a factor of 20.