Semi-classical Expansion of the Smooth Part of the Density of States, with Application to the Hydrogen Atom in Uniform Magnetic Field

We give a method to compute the smooth part of the density of states in a semi-classical expansion, when the Hamiltonian contains a Coulomb potential and non-cartesian coordinates are appropriate. We apply this method to the case of the hydrogen atom in a magnetic field with fixed $z$-component of the angular momentum. This is then compared with numerical results obtained by a high precision finite element approach. The agreement is excellent especially in the \emph{chaotic} region of the spectrum. The need to go beyond the Thomas-Fermi model is clearly established.