Two realizations of Schrِdinger operators on Riemannian manifolds

We consider a Schrödinger differential expression P = Δ M + V on a complete Riemannian manifold ( M , g ) with metric g, where Δ M is the scalar Laplacian on M and V is a real-valued locally integrable function on M. We study two self-adjoint realizations of P in L 2 ( M ) and show their equality. This is an extension of a result of S. Agmon.