Weyl Expansion for Symmetric Potentials

Abstract

We present a semiclassical expansion of the smooth part of the density of states in potentials with some form of symmetry. The density of states of each irreducible representation is separately evaluated using the Wigner transforms of the projection operators. For discrete symmetries the expansion yields a formally exact but asymptotic series in $\hbar$, while for the rotational $SO(n)$ symmetries the expansion requires averaging over angular momentum as well as energy. A numerical example is given in two dimensions, in which we calculate the leading terms of the Weyl expansion as well as the leading periodic orbit contributions to the symmetry reduced level density.