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

We consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than E for elliptic operators in L2(Rd ). 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. If the volume of this region is O(h), then we obtain remainder estimates O(h1−d ) with no assumptions on the Hessian of the principal symbol at the critical level. Moreover we do not assume that the coefficients are smooth—all results hold if second order derivatives of coefficients are Hölder continuous. © 2007 Elsevier Inc. All rights reserved.