Subprincipal terms in Szegö estimates
type: Item
instance of: scholarly article
title: Subprincipal terms in Szegö estimates
zbMATH Open document ID: 0881.58064
author: K. Okikiolu
author: Victor Guillemin
published in: Mathematical Research Letters
publication date: 1998-02-26T00:00:00Z
review text: The authors consider a first order positive self-adjoint elliptic pseudo-differential operator \(Q\) (operating on half-densities) defined on a compact \(d\)-dimensional manifold \(X\), whose principal symbol \(q\) defined as a function on \(T^*X \backslash \{0\}\) satisfies \(q(x,\xi)= q(x,-\xi)\) and whose subprincipal manifold is zero. Let \(\lambda_i\) be the eigenvalues of \(Q\) and \(P_\lambda\) be the orthogonal projection onto the space spanned by the eigenvectors corresponding to \(\lambda_i <\lambda\). If \(A\) is a zeroth order pseudo-differential operator, the authors analyze the behavior as \(\lambda\to +\infty\) of \(\text{Tr} P_\lambda AP_\lambda\) and announce, under the condition that the set of periodic points for the Hamiltonian flow associated to \(H_q\) is of measure zero in \(T^*X \backslash \{0\}\), a two terms asymptotics improving the classical Weyl formula. More generally, they also consider, for \(k\in \mathbb{N}\), \(\text{Tr} (P_\lambda A P_\lambda)^k\) and other spectral quantities.
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1019795
zbMATH Keywords: first order positive selfadjoint elliptid pseudo-differential operator
zbMATH Keywords: principal symbol
zbMATH Keywords: subprincipal manifold
zbMATH Keywords: eigenvalue
zbMATH Keywords: periodic points
zbMATH Keywords: Hamiltonian flow
zbMATH Keywords: asymptotics
zbMATH Keywords: Weyl formula
reviewed by: Bernard Helffer
MaRDI profile type: Publication
DOI: 10.4310/MRL.1997.V4.N1.A15
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description: scientific article; zbMATH DE number 1019795