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

Article type: Research Article

Authors: Zielinski, Lech

Affiliations: LMPA, Centre Mi-Voix, Université du Littoral, B.P. 699, 62228 Calais Cedex, France E-mail: Lech.Zielinski@lmpa.univ-littoral.fr

Abstract: 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(h1−d) for a class of symbols with critical points and non-smooth coefficients.

Keywords: semiclassical approximation, eigenvalue asymptotics, critical energy

Journal: Asymptotic Analysis, vol. 53, no. 1-2, pp. 97-123, 2007