# Quantum monodromy in prolate ellipsoidal billiards

preprint

posted on 2006-01-24, 11:25 authored by Holger Waalkens, Holger R. DullinThis is the third in a series of three papers on quantum billiards with elliptic and ellipsoidal
boundaries. In the present paper we show that the integrable billiard inside a prolate
ellipsoid has an isolated singular point in its bifurcation diagram and, therefore, exhibits
classical and quantum monodromy. We derive the monodromy matrix from the requirement
of smoothness for the action variables for zero angular momentum. The smoothing procedure
is illustrated in terms of energy surfaces in action space including the corresponding smooth
frequency map. The spectrum of the quantum billiard is computed numerically and the
expected change in the basis of the lattice of quantum states is found. The monodromy
is already present in the corresponding two-dimensional billiard map. However, the full
three degrees of freedom billiard is considered as the system of greater relevance to physics.
Therefore, the monodromy is discussed as a truly three-dimensional e ect.

## History

## School

- Science

## Department

- Mathematical Sciences

## Pages

3170340 bytes## Publication date

2001## Notes

This is a pre-print. The definitive version: WAALKENS, H. and DULLIN, H.R., 2002. Quantum monodromy in prolate ellipsoidal billiards. Annals of Physics, 295(1), pp.81-112, is available at: http://www.sciencedirect.com/science/journal/00034916.## Language

- en