On the essential spectrum of phase-space anisotropic pseudodifferential operators

Abstract

AbstractA phase-space anisotropic operator in=L2(ℝn) is a self-adjoint operator whose resolvent family belongs to a naturalC*-completion of the space of Hörmander symbols of order zero. Equivalently, each member of the resolvent family is norm-continuous under conjugation with the Schrödinger unitary representation of the Heisenberg group. The essential spectrum of such a phase-space anisotropic operator is the closure of the union of usual spectra of all its “phase-space asymptotic localizations”, obtained as limits over diverging ultrafilters of ℝn×ℝn-translations of the operator. The result extends previous analysis of the purely configurational anisotropic operators, for which only the behavior at infinity in ℝnwas allowed to be non-trivial.