Quantum states in a champagne bottle

Features of the classical mechanics under the potential are related to the quantum-mechanical spectrum. A known topological obstruction to the existence of global action-angle variables is shown to be reflected in a sharp change in the disposition of eigenvalues around E = 0. The energies of states labelled by a radial quantum number v vary smoothly with the angular momentum for E<0 but have a discontinuous derivative at for E>0 . An alternative labelling , where shows the opposite effect. Semiclassical arguments based on the forms of positive and negative energy trajectories also suggest a near symmetry between the positive second derivative ( at energy and the negative counterpart ( at energy , which is confirmed in the quantum-mechanical spectrum. Minor deficiencies in the Bohr-Sommerfeld rule are removed by introducing an analytical phase correction for states with and .