Comparing periodic-orbit theory to perturbation theory in the asymmetric infinite square well

TL;DR

This paper compares periodic-orbit theory and perturbation theory in analyzing the asymmetric infinite square well, showing that periodic-orbit theory provides more accurate energy estimates outside the perturbation convergence limit.

Contribution

It demonstrates the effectiveness of periodic-orbit theory over perturbation theory for certain energy regimes in a simple quantum system with a potential step.

Findings

Periodic-orbit theory aligns with perturbation theory within the convergence limit.

Periodic-orbit theory outperforms perturbation theory for energies above the potential step.

Non-Newtonian orbits significantly influence second-order perturbation corrections.

Abstract

An infinite square well with a discontinuous step is one of the simplest systems to exhibit non-Newtonian ray-splitting periodic orbits in the semiclassical limit. This system is analyzed using both time-independent perturbation theory (PT) and periodic-orbit theory and the approximate formulas for the energy eigenvalues derived from these two approaches are compared. The periodic orbits of the system can be divided into classes according to how many times they reflect from the potential step. Different classes of orbits contribute to different orders of PT. The dominant term in the second-order PT correction is due to non-Newtonian orbits that reflect from the step exactly once. In the limit in which PT converges the periodic-orbit theory results agree with those of PT, but outside of this limit the periodic-orbit theory gives much more accurate results for energies above the potential step.