 Semiclassical density of states in a central potential

For a spinless nonrelativistic particle in a central potential, $g(ϵ)=(2m{)}^{3/2}(4{\pi }^2{\hslash }^3{)}^{-1}{\int }_{U<ϵ}\sqrt{ϵ-U}{d}^{3}r$. Spherical symmetry reduces this to a radial integral over every allowed shell. It is a leading phase-space approximation.