Quantum Billiards on Polyhedral Surfaces

Abstract

Solutions to Schrodinger's equation for free particles confined to the surface of a regular polyhedron can be determined by solving the same equation on an unfolding of the polyhedral surface embedded in $\mathbb{R}^2$ and enforcing appropriate boundary conditions to ensure that the wavefunction is well-defined and smooth. Previous work by Bělín et al has determined the complete set of eigenmodes for particles on a tetrahedron, all of which are solvable. We extend their work to particles confined to the surfaces of the triangle- and square-faced regular polyhedra as well as square and rectangular prisms. In all cases aside from the tetrahedron, we find that the solvable eigenmodes are incomplete, and conjecture that the remaining eigenmodes are unsolvable and quantum chaotic. To support this conjecture, we examine the energy level spacing statistics of numerically-computed energy eigenmodes for a tight-binding model lattice on the polyhedral surface. Lastly, we consider the classical limit of the system—billiards on polyhedral surfaces—and show that the classical system is integrable for the tetrahedron but pseudo-integrable for all other polyhedra under consideration, consistent with the quantum result.