Subprincipal terms in Szegö estimates
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.
first order positive selfadjoint elliptid pseudo-differential operator
scientific article; zbMATH DE number 1019795
scientific article