Localization and Semibounded Energy - A Weak Unique Continuation Theorem

TL;DR

This paper proves a weak unique continuation property for self-adjoint Dirac-type operators on closed manifolds, showing that local vanishing implies global vanishing for certain invariant subspaces.

Contribution

It establishes a weak unique continuation theorem for semibounded restrictions of Dirac-type operators on closed Riemannian manifolds.

Findings

Elements in D-invariant subspaces with semibounded restriction vanish everywhere if they vanish locally.

The result extends unique continuation properties to a broader class of operators and subspaces.

Provides a theoretical foundation for localization phenomena in geometric analysis.

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 u in H has the weak unique continuation property, i.e. if u vanishes on a nonempty open subset of M, then it vanishes on all of M.