ASYMPTOTIC DISTRIBUTION OF EIGENVALUES FOR SOME ELLIPTIC OPERATORS WITH SIMPLE REMAINDER ESTIMATES

Abstract

Abstract. We are interested in remainder estimates in the Weyl formula for the asymptotic number of eigenvalues of certain elliptic operators on Rd and on a smooth compact manifold without boundary. The main aim of this paper is to compare spectral asymptotics of operators with irregular coefficients and certain classes of smoothed operators for which the Weyl formula is derived by means of elementary pseudodifferential calculus. The remainder estimates are obtained here essentially with an exponent less than one half of the optimal exponent known in the case of smooth coef-ficients. The presentation is self-contained (we do not require any knowledge of the subject) and will be continued in a subsequent paper, where sharper remainder estimates will be proved.