Friedrichs Extension and Min–Max Principle for Operators with a Gap

  • Lukas SchimmerEmail author
  • Jan Philip Solovej
  • Sabiha Tokus
Original Paper


Semibounded symmetric operators have a distinguished self-adjoint extension, the Friedrichs extension. The eigenvalues of the Friedrichs extension are given by a variational principle that involves only the domain of the symmetric operator. Although Dirac operators describing relativistic particles are not semibounded, the Dirac operator with Coulomb potential is known to have a distinguished extension. Similarly, for Dirac-type operators on manifolds with a boundary a distinguished self-adjoint extension is characterised by the Atiyah–Patodi–Singer boundary condition. In this paper, we relate these extensions to a generalisation of the Friedrichs extension to the setting of operators satisfying a gap condition. In addition, we prove, in the general setting, that the eigenvalues of this extension are also given by a variational principle that involves only the domain of the symmetric operator. We also clarify what we believe to be inaccuracies in the existing literature.

Mathematics Subject Classification

49R05 49S05 47B25 81Q10 



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Copyright information

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Authors and Affiliations

  1. 1.QMATH Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

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