On self-adjointness of a Schrödinger operator on differential forms

Abstract

Let M M be a complete Riemannian manifold and let Ω ∙ ( M ) \Omega ^{\bullet }(M) denote the space of differential forms on M M . Let d : Ω ∙ ( M ) → Ω ∙ + 1 ( M ) d:\Omega ^{\bullet }(M)\to \Omega ^{\bullet +1}(M) be the exterior differential operator and let Δ = d d ∗ + d ∗ d \Delta =dd^{*}+d^{*}d be the Laplacian. We establish a sufficient condition for the Schrödinger operator H = Δ + V ( x ) H=\Delta +V(x) (where the potential V ( x ) : Ω ∙ ( M ) → Ω ∙ ( M ) V(x):\Omega ^{\bullet }(M)\to \Omega ^{\bullet }(M) is a zero order differential operator) to be self-adjoint. Our result generalizes a theorem by I. Oleinik about self-adjointness of a Schrödinger operator which acts on the space of scalar valued functions.