Sharp semiclassical estimates for the number of eigenvalues below a degenerate critical level

We consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than E for elliptic operators in $L^{2}(\mathbb{R}^{d})$ . We describe a method of obtaining remainder estimates related to the volume of the region of the phase space in which the principal symbol takes values belonging to the intervals [E′;E′+h], where E′ is close to E. This method allows us to derive sharp remainder estimates O(h^1−d) for a class of symbols with critical points and non-smooth coefficients.