Self-adjointness of certain second order differential operators on Riemannian manifolds

Abstract

where u E C,“(R”) and Dj = (a/a X, ib,(x)). The problem of its essential self-adjointness (i.e., “when does L have a unique self-adjoint extension ?“) has been widely investigated. When the context is changed from Euclidean space to a more general differentiable manifold, however, there is a paucity of results. The salient theorems in the latter setting are those of Gaffney [7] in 1951 on the Laplace-Beltrami operator on a complete Riemannian manifold, and Cordes [5] in 1972 on its powers. Most recently Chernoff [4] has obtained a result on a type of general second order elliptic operator with singular potential. The reason that self-adjointness theorems on manifolds are hard to come by is that is usually no single set of coefficient functions that describes the operator. One is only given such functions locally. Generally speaking this difficulty must be handled in one of two ways. The first is by means of global (i.e., coordinate free) invariants associated with the operator. The second is to argue and estimate locally in such a way ultimately things can be “pieced together” into a global and workable result. Using each of these approaches we extend to complete Riemannian manifolds two recent results on Euclidean space of A. Devinatz [6]. In the first, the potential is assumed to be in Li,, and bounded below locally with decay to minus infinity tied to a quantity whose Euclidean formulation is 1 x i -2 x a,,(~)lr,x, . In the manifold setting this quantity becomes a(~; dr(x)), where a(x, V) is the symbol of the operator and dr(zc) is the differential of a general distance function. The key to the proof is Proposition 3.1 in which we establish certain global smoothness