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arXiv · 1806.05206

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

Lukas Schimmer·Jan Philip Solovej·Sabiha Tokus
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Abstract

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.

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Lukas Schimmer, Jan Philip Solovej, Sabiha Tokus. 2018-06-13. Friedrichs Extension and Min-Max Principle for Operators with a Gap. https://arxiv.org/abs/1806.05206

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