Localization and semibounded energy — a weak unique continuation theorem

Abstract

Let D be a self-adjoint differential operator of Dirac type acting on sections in a vector bundle over a closed Riemannian manifold M. Let H be a closed D-invariant subspace of the Hilbert space of square integrable sections. Suppose D restricted to H is semibounded. We show that every element ψ∈ H has the weak unique continuation property, i.e. if ψ vanishes on a nonempty open subset of M, then it vanishes on all of M.