Sharp asymptotics in Weyl's law

• Main Author: Taherkhani, Farnaz.
• Format: Others
• Language: en
• Published: McGill University 2006
• Subjects: Mathematics.
• Online Access: http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=99208

Let (M, g) be a closed n-dimensional Riemannian manifold with metric g and Laplace-Beltrami operator Delta. Let 0 = l20 < l21 < ... be the eigenvalues of Delta. For the spectral counting function N(t) = #{j, lambda j ≤ t}, we give a detailed proof of Hormander's theorem states that: Nt=vol BnvolM 2pn tn/2+Ot n-1/2, where vol(Bn) is the volume of the n-dimensional unit ball and by O(t(n -1)/2) we mean a term which grows no faster than Ct (n-1)/2 as t tends to infinity. O(t (n-1)/2) is the sharp error term estimate in Weyl's law. (Actually we get the estimate of O(t(n -1)/m) for a differential operator of order m). We also obtain an off diagonal estimate for the remainder term is a generalization of Weyl's law.