POSITIVE PERTURBATIONS OF SELF-ADJOINT SCHRÖDINGER OPERATORS ON RIEMANNIAN MANIFOLDS

Abstract. We consider a Schrödinger differential expression L0 = ∆M+V0 on a (not necessar-ily complete) Riemannian manifold (M, g) with metric g, where ∆M is the scalar Laplacian on M and V0 is a real-valued locally square integrable function on M. We consider a perturbation L0 + V, where V is a non-negative locally square-integrable function on M, and give sufficient conditions for L0 + V to be essentially self-adjoint on C c (M). This is an extension of a re-sult of T. Kappeler. The proof adopts Kappeler’s technique, but requires the use of positivity preserving property of resolvents of certain self-adjoint operators in L2(M)