The form sum and the Friedrichs extension of Schrödinger-type operators on Riemannian manifolds

Abstract

We consider H V = Δ M + V, where (M, g) is a Riemannian manifold (not necessarily complete), and Δ M is the scalar Laplacian on M. We assume that V = V 0 + V 1 , where V 0 ∈ L 2 loc(M) and -C ≤ V 1 ∈ L 1 loc(M) (C is a constant) are real-valued, and Δ M + V 0 is semibounded below on C∞c(M). Let T 0 be the Friedrichs extension of (Δ M + V 0 )|C∞c(M). We prove that the form sum T 0 +(V 1 coincides with the self-adjoint operator T F associated to the closure of the restriction to C∞ c(M) x C∞c(M) of the sum of two closed quadratic forms of To and V 1 . This is an extension of a result of Cycon. The proof adopts the scheme of Cycon, but requires the use of a more general version of Kato's inequality for operators on Riemannian manifolds.