Journal of Operator Theory

Volume 39, Issue 2, Spring 1998 pp. 249-282.

Asymptotic distribution of eigenvalues for some elliptic operators with simple remainder estimates

Authors: Lech Zielinski
Author institution: Institut de Maths de Paris - Jussieu, UMR 9994, Universite Paris 7 Denis Diderot, case 7012, 2 Place Jussieu, 75252 Paris Cedex 05, France

Summary: We are interested in remainder estimates in the Weyl formula for the asymptotic number of eigenvalues of certain elliptic operators on ${\bbb R}^d$ 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 coefficients. 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.

Keywords: Spectral asymptotics, Weyl formula, self-adjoint elliptic operators with irregular coefficients, remainder estimates

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