Semiclassical Spectral Asymptotics on Foliated Manifolds

Abstract

We consider a (hypo)elliptic pseudodifferential operator Ah on a closed foliated manifold (M,ℱ), depending on a parameterh > 0, of the form Ah = A+hmB, where A is a formally self–adjoint tangentially elliptic operator of orderμ > 0 with the nonnegative principal symbol and B is a formally self–adjoint classical pseudodi.erential operator of orderm > 0 on M with the holonomy invariant transversal principal symbol such that its principal symbol is positive, if μ < m, and its transversal principal symbol is positive, if μ ≥ m. We prove an asymptotic formula for the eigenvalue distribution function Nh(λ) of the operator Ah when h tends to 0 and λ is constant.