DISCRETE SYMMETRIES IN THE WEYL EXPANSION FOR QUANTUM BILLIARDS

We consider two- and three-dimensional quantum billiards with discrete symmetries. The boundary condition is either Dirichlet or Neumann. We derive the first terms of the Weyl expansion for the level density projected onto the irreducible representations of the symmetry group. The formulae require only knowledge of the character table of the group and the geometrical properties (such as surface, perimeter etc ... ) of sub-parts of the billiard invariant under a group transformation. As an illustration, the method is applied to the icosahedral billiard.