Abstract
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.
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Milatovic, O., Truc, F. Self-adjoint extensions of differential operators on Riemannian manifolds. Ann Glob Anal Geom 49, 87–103 (2016). https://doi.org/10.1007/s10455-015-9482-0
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DOI: https://doi.org/10.1007/s10455-015-9482-0
Keywords
- Essential self-adjointness
- Hermitian vector bundle
- Higher-order differential operator
- Riemannian manifold