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

TY - JOUR

T1 - Friedrichs Extension and Min–Max Principle for Operators with a Gap

AU - Schimmer, Lukas

AU - Solovej, Jan Philip

AU - Tokus, Sabiha

PY - 2020/2/1

Y1 - 2020/2/1

N2 - 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.

AB - 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.

KW - math-ph

KW - math.MP

KW - math.SP

KW - 49R05, 49S05, 47B25, 81Q10, 58J32

U2 - 10.1007/s00023-019-00855-7

DO - 10.1007/s00023-019-00855-7

M3 - Journal article

SN - 1424-0637

JO - Annales Henri Poincare

JF - Annales Henri Poincare

ER -