Essential Self-adjointness of Differential Operators on Riemannian Manifolds

In this paper we have studied the essential self-adjointness for the di⁄erential operator of the form : T = (cid:1) 8 + V; on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential V satisfying a bound from below by a non-positive function depending on the distance from a point. We give su¢cient condition for the essential self-adjointness of such di⁄erential operator on Riemannian Manifolds.


Introduction
The study of self-adjointness of di¤erential operators on Euclidean spaces has many works, such as [8] and [13]. M. Ga¤ney initiated this problem on Riemannian manifolds in [8]. This work focused on the essential self-adjointness of the scalar Laplacian and the Hodge Laplacian. H. Cordes proved the case of positive integer powers of the scalar Laplacian and Hodge Laplacian in [5]. Generalisations to the case of essential self-adjointness of positive integer powers of the …rst order di¤erential operators was proved by P. Cherno¤ in [4]. After these works many studied of the essential self-adjointness problem on Riemannian manifolds were done such as [6], [8] and [13]. Perturbations were considered of a biharmonic operator, 2 + V , over a complete Riemannian manifold M by Milatovic in [11], where V 2 L 1 loc (End E) is the potential, End E denotes the endomorphism bundle associated to E and V satis…es a bound from below by a non-positive function depending on the distance from a point. Milatovic was obtained suitable localised derivative estimates, which was important for the proof of the essential self-adjointness of operators on C 1 c (E). For a study of separation in the context of a perturbation of the magnetic Bi-Laplacian on L 2 (M ), see the paper [1]. Atia studied the separation problem on Riemannian manifolds in [2] and [3]. In this paper, we consider the operator, 8 + V , acting on sections of a Hermitian vector bundle E over a complete Riemannian manifold M , where = r + r denotes a Bochner Laplacian associated to a Hermitian connection r, V is a potential satis…ed that V 2 L 1 loc (End E), and satis…es a bound from below by a non-positive function depending on the distance from a point. Let (M; g) be a smooth connected Riemannian n-manifold without boundary, where g is the Riemannian metric on M . We de…ne the following components: M denotes the Laplace Beltrami operator on functions on M , the associated Riemannian volume form by d , also r M denotes the canonical Levi-Civita connection on M , and the associated curvature tensor by R m . The pair (E; h) is the smooth Hermitian vector bundle over M , where h is Hermitian metric. r denotes the metric connection on E, this connection gives a curvature tensor F . The formal adjoint of r will be denoted by r + , with the associated Bochner Laplacian being given by := r + r. We de…ne C 1 (M ) and C 1 c (M ) are the smooth functions and smooth functions with compact support on M respectively. Similarly, We de…ne C 1 (E) and C 1 c (E) are the smooth sections and smooth sections with compact support of E respectively. We will use the notation L 2 (E) to denote the Hilbert space of square integrable sections of E, with the inner product We will denote the associated L 2 -norm by where juj 2 = h (u; u). We denote local Sobolev spaces of sections in L 2 (E), by W k;2 loc (E), with k indicating the highest order of derivatives. Let the distance from a point, which we denote by r, there exist a …xing point x 2 M we let where d is the distance function induced from the Riemannian metric g on M , for all x 2 M .

Bounded Geometry
De…nition 1 Let (M; g) be a smooth non-compact Riemannian manifold. We say (M; g) admits bounded geometry if the following conditions are satis…ed.
(1) r m > 0; (2) sup x2M r k R m (x) C k for k 0, and C k > 0, where r m denotes the injectivity radius of M , r is the Levi-Civita connection, and R m denotes the curvature tensor.
De…nition 2 Let M be a smooth manifold and (E; h; r) a Hermitian vector bundle over M , with Hermitian metric h and connection r, we say the triple (E; h; r) admits k-bounded geometry if the following condition is satis…ed: sup x2M r j F (x) C j for 0 j k, and C j > 0, where F is the curvature tensor associated to r. We say (E; h; r) admits bounded geometry if it admits k-bounded geometry for all k 0. In this paper we let two conditions on the geometry of our Riemannian manifolds and the vector bundles over them.
All Riemannian manifolds (M; g) are admit bounded geometry. All Hermitian vector bundles (E; h; r) are admit 1-bounded geometry.

Distance functions
In this paper we will be using distance functions. We de…ne . From the second condition of the previous lemma we have where is a multi-index, C > 0 is a constant, and the derivative @ y is taken with respect to normal coordinates. In particular, this implies to Let (M; g) be a Riemannian manifold and (E; h; r) be a Hermitian vector bundle over M , with Hermitian metric h and metric connection r. Let u 2 Also we have for the product We will be iterating, we obtain Finally, we have We will be using lemma 5:15 in [9].
Lemma 5 Let E be a Hermitian vector bundle over a Riemannian manifold (M; g), with metric compatible connection r, let = r + r denote the Bochner Laplacian, and let u be a section of E. We have We will be applying the previous lemma for n = k = 1.
Corollary 6 Let E be a Hermitian vector bundle over a Riemannian manifold (M; g), with metric compatible connection r, let = r + r denote the Bochner Laplacian, and let u be a section of E. We have We will use localised derivative estimates needed for the proof of the next theorem. We will need the following result of Saratchandran, H.
Proposition 8 For u 2 W 2;2 loc (E) and > 0 su¢ciently small, we have the following estimate x k 2 u 2 : Proof. We have so we obtain applying Cauchy-Schwarz inequality, we get where to get the …rst inequality, we have used our bounded geometry conditions 4 and 5. We can estimate the term x k r 3 u 2 by using proposition 9.
x k r 3 u We also estimate the term x k 1 r 3 u 2 by using proposition 9.
From (8), we have the following formula for the Laplacian of the cut-o¤ function x Corollary 9 We have the following estimate M x 2k C x 2k 2 for some constant C > 0.
In order to get G 3 , applying d to the above formula for M x 2k we get We will de…ne the minimal operator associated to T by T min u := T u with domain D min := C 1 c (E). We also de…ne the maximal operator associated to T as the adjoint of the minimal operator, T max := (T min ) , since for a linear densely de…ned operator L, since L denote the adjoint. We can be de…ned the domain of the operator T max as where T max u := T u for u 2 D max .
The following lemma can be existed as a Bilaplacian version of Milatovic ; s lemma 4.1 in [11].
Lemma 10 Let V satis…es the hypotheses of the following theorem, assume u 2 Dom (T max ) and T max u = i u, for any 2 R. Then given > 0 su¢ciently small, we get the following estimate where C 1 ( ) and C 2 ( ) are constants depending on such that lim !0 C 1 ( ) < 1 and lim !0 C 2 ( ) < 1.