zbMATH — the first resource for mathematics

Non Weylian spectral asymptotics with accurate remainder estimate. (English) Zbl 0761.47029
Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math., Palaiseau 1990-1991, No.V, 10 p. (1991).
This paper is introduced by the author as a presentation at seminars at Ecole Polytechnique, Palaiseau, France, during 1990-1991, and involves second order differential operators of the form \[ A=\sum_{j,k} D_ j g^{jk}(x)D_ k+\sum_ j(b_ j(x)D_ j+D_ j b_ j(x))+c(x), \] where \(g^{jk}=g^{kj}\), \(b_ j,c\in C^ K\) are real-valued and \(A\) is a uniformly elliptic operator, so that \(\sum_{j,k} g^{jk} n_ j n_ k\geq| n|^ 2/c\) \(\forall n\in R^ d\). If \(N(t)\) represents the eigenvalue counting function of \(A\), then the main results of the paper involve expressions for \(N(t)\) in the form \(N(t)=N_ 0^ w(t)+N^{w'}(t)+O(R)\), where \(R\) is expressible in terms of \(t^{{1\over2}(d-\ell)}\) or \(t^{{1\over2}(d-\ell)}\log t\), and \(N_ 0^ w(t)\), \(N_ 1^{w'}(t)\) are expressible in terms of Weylian integral expressions. Earlier results which do not include the term \(N_ 1^{w'}(t)\) have been considered by the author and S. I. Fedorova [Funct. Anal. Appl. 20, No. 4, 277-281 (1986; Zbl 0628.35077)].
47G30 Pseudodifferential operators
35P20 Asymptotic distributions of eigenvalues in context of PDEs
Zbl 0628.35077
Full Text: Numdam EuDML