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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)].
##### MSC:
 47G30 Pseudodifferential operators 35P20 Asymptotic distributions of eigenvalues in context of PDEs
Zbl 0628.35077
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