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Self-adjoint extensions of differential operators on Riemannian manifolds

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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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Acknowledgments

We are grateful to the referee for useful suggestions and comments.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL, 32224, USA

    Ognjen Milatovic

  2. Unité mixte de recherche CNRS-UJF 5582, Institut Fourier, Grenoble University, BP 74, 38402, Saint Martin d’Hères Cedex, France

    Françoise Truc

Authors
  1. Ognjen Milatovic
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  2. Françoise Truc
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Correspondence to Ognjen Milatovic.

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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

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