Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues

  • Harold Donnelly
  • Nicola Garofalo
Article

DOI: 10.1007/BF02921722

Cite this article as:
Donnelly, H. & Garofalo, N. J Geom Anal (1997) 7: 241. doi:10.1007/BF02921722

Abstract

Suppose thatMn is a complete, noncompact, Riemannian manifold. If Δ denotes the Laplace operator ofM, one has associated Schrödinger operators − Δ +V. Conditions onV are formulated, which ensures the essential self-adjointness of − Δ +V. In particular, ifV ∈ Qα,loc (Mn), the local Stummel class, andV ≥ − c outside of a compact set, then − Δ +V is essentially self-adjoint on C0(Mn). In addition, essential self-adjointness is proved for potentials which are strongly singular at a point. The absence of eigenvalues of −Δ +V is also studied. This relies upon Rellich-type identities. The results on strongly singular potentials make use of a generalization of the classical uncertainty principle, inRn, to Riemannian manifolds with a pole.

Math Subject Classification

58G0358G25

Key Words and Phrases

Riemannian manifoldLaplacianstrongly singular potentialsRellich identitiesuncertainty principle

Copyright information

© Mathematica Josephina, Inc. 1997

Authors and Affiliations

  • Harold Donnelly
    • 1
  • Nicola Garofalo
    • 1
  1. 1.Department of MathematicsPurdue UniversityWest Lafayette