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

TL;DR

This paper extends the Friedrichs extension concept to operators with a spectral gap, establishing a variational principle for eigenvalues that applies to a broader class of operators, including Dirac operators.

Contribution

It generalizes the Friedrichs extension and variational eigenvalue characterization to operators satisfying a gap condition, encompassing Dirac operators with boundary conditions.

Findings

Eigenvalues of the generalized extension are characterized by a variational principle.

The extension applies to operators with a spectral gap, including Dirac operators.

The results unify and extend known spectral theories for semibounded and Dirac operators.

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.