Mathematische Zeitschrift

, Volume 275, Issue 1–2, pp 331–348 | Cite as

Path integrals and the essential self-adjointness of differential operators on noncompact manifolds

  • Batu GüneysuEmail author
  • Olaf Post


We consider Schrödinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on \(\mathsf{C }^{\infty }_0\), and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli–Dirac operators that describe the energy of Hydrogen type atoms on Riemannian \(3\)-manifolds.


Vector Bundle Dirac Operator Operator Core Noncompact Riemannian Manifold Clifford Multiplication 
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The first author (BG) is indebted to Ognjen Milatovic for many discussions on essential self-adjointness in the past three years, in particular, for bringing the reference [13] into our attention (which helped us to remove an unnecessary assumption from the original version of Theorem 1.1). Both authors kindly acknowledge the financial support given by the SFB 647 “Space–Time–Matter” at the Humboldt University Berlin, where this work has been started.


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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut für Mathematik, Humboldt-Universität zuBerlinGermany
  2. 2.School of MathematicsCardiff UniversityWalesUK

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