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We give a method to compute the smooth part of the density of states in asemi-classical expansion, when the Hamiltonian contains a Coulomb potential andnon-cartesian coordinates are appropriate. We apply this method to the case ofthe hydrogen atom in a magnetic field with fixed $z$-component of the angularmomentum. This is then compared with numerical results obtained by a highprecision finite element approach. The agreement is excellent especially in the\emph{chaotic} region of the spectrum. The need to go beyond the Thomas-Fermimodel is clearly established.Comment: 12 pages, 3 figure

Semiclassical expansion of the smooth part of the density of states, with application to the hydrogen atom in a uniform magnetic field