Self-adjoint extensions of differential operators on Riemannian manifolds

Ognjen Milatovic (1) , Francoise Truc (2)

1 UNF - University of North Florida [Jacksonville]
2 IF - Institut Fourier

We study $H=D^*D+V$, where $D$ is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold $M$, and $V$ is a Hermitian bundle endomorphism. In the case when $M$ is geodesically complete, we establish the essential self-adjointness of positive integer powers of $H$. In the case when $M$ is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of $H$, expressed in terms of the behavior of $V$ relative to the Cauchy boundary of $M$.

Mots clés: essential self-adjointness; Schrödinger-type operator; Riemannian manifold; vector bundle

Domaines: Théorie spectrale [math.SP]; Physique mathématique [math-ph]

Ognjen Milatovic, Francoise Truc. Self-adjoint extensions of differential operators on Riemannian manifolds. Annals of Global Analysis and Geometry, 2016, 49 (1), pp.87-103. ⟨10.1007/s10455-015-9482-0⟩. ⟨hal-01149461⟩