Spectral properties of zeroth-order pseudodifferential operators

Abstract

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, { E S } S∈ R as a bounded operator on L 2( X, dx). Using theorems of Weyl ( Rend. Circ. Mat. Palermo, 27 (1909) , 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec ESS Q = range( q( x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family { E S } S∈ R , it is shown that if dE ds is considered as a distribution on R × X× X it is in fact a Lagrangian distribution near the set {σ=0}⊂ T ∗( R×X×X) 0 where ( s, x, y, σ, ξ, η) are coordinates on T ∗( R × X× X) induced by the coordinates ( s, x, y) on R × X× X. This leads to an easy proof that ƒ(Q) is a pseudodifferential operator if ƒ∈ C ∞( R ) and to some results on the microlocal character of E s . Finally, a look at the wavefront set of dE ds leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q( x, ξ).