Annales Henri Poincaré

, Volume 16, Issue 10, pp 2367–2397 | Cite as

On Self-Adjoint Extensions and Symmetries in Quantum Mechanics

  • Alberto Ibort
  • Fernando LledóEmail author
  • Juan Manuel Pérez-Pardo


Given a unitary representation of a Lie group G on a Hilbert space \({\mathcal H}\), we develop the theory of G-invariant self-adjoint extensions of symmetric operators using both von Neumann’s theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the G-invariant unbounded operator. We also prove a G-invariant version of the representation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of G-invariant self-adjoint extensions of the Laplace–Beltrami operator on a smooth Riemannian manifold with boundary on which the group G acts. These extensions are labeled by admissible unitaries U acting on the L 2-space at the boundary and having spectral gap at −1. It is shown that if the unitary representation V of the symmetry group G is traceable, then the self-adjoint extension of the Laplace–Beltrami operator determined by U is G-invariant if U and V commute at the boundary. Various significant examples are discussed at the end.


Hilbert Space Quadratic Form Unitary Representation Symmetric Operator Representation Versus 
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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Alberto Ibort
    • 1
    • 2
  • Fernando Lledó
    • 1
    • 2
    Email author
  • Juan Manuel Pérez-Pardo
    • 1
    • 2
    • 3
  1. 1.Department of MathematicsUniversidad Carlos III de MadridLeganés (Madrid)Spain
  2. 2.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain
  3. 3.INFN-Sezione di NapoliNaplesItaly

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