Comparison of classical and quantum spectra for a totally bound potential

Abstract

The quantal energy spectrum is compared with the classical motion for the totally bound potential 1/2(x2+y2)+ax2y2. The classical phase space is filled with regular trajectories at lower energies, but as the energy is increased both regular and irregular trajectories are observed to coexist. At very high energies the classical phase space is almost totally filled with irregular trajectories. The work reported here is similar to that performed by the authors on the Henon-Heiles potential with the purpose of testing the prediction by Percival (1973) that there is good agreement between the amount of classical irregular motion and the portion of energy eigenvalues sensitive to small changes in the perturbation parameter. However, the potential investigated has several computational advantages over the Henon-Heiles potential as well as avoiding complications due to quantum mechanical tunnelling. The results show good agreement with Percival's predictions.