Friedrichs Extension and Min–Max Principle for Operators with a Gap
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Friedrichs Extension and Min–Max Principle for Operators with a Gap. / Schimmer, Lukas; Solovej, Jan Philip; Tokus, Sabiha.
In: Annales Henri Poincare, Vol. 21, No. 2, 01.02.2020, p. 327-357.Research output: Contribution to journal › Journal article › Research › peer-review
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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-PatodiSinger 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-PatodiSinger 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.
U2 - 10.1007/s00023-019-00855-7
DO - 10.1007/s00023-019-00855-7
M3 - Journal article
VL - 21
SP - 327
EP - 357
JO - Annales Henri Poincare
JF - Annales Henri Poincare
SN - 1424-0637
IS - 2
ER -
ID: 236317096