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An infinite square well with a discontinuous step is one of the simplestsystems to exhibit non-Newtonian ray-splitting periodic orbits in thesemiclassical limit. This system is analyzed using both time-independentperturbation theory (PT) and periodic-orbit theory and the approximate formulasfor the energy eigenvalues derived from these two approaches are compared. Theperiodic orbits of the system can be divided into classes according to how manytimes they reflect from the potential step. Different classes of orbitscontribute to different orders of PT. The dominant term in the second-order PTcorrection is due to non-Newtonian orbits that reflect from the step exactlyonce. In the limit in which PT converges the periodic-orbit theory resultsagree with those of PT, but outside of this limit the periodic-orbit theorygives much more accurate results for energies above the potential step.

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