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Friedrichs extension and min–max principle for operators with a gap

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journal contribution
posted on 2023-11-21, 17:13 authored by Lukas SchimmerLukas Schimmer, Jan Philip Solovej, Sabiha Tokus
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.

Funding

The Mathematics of the Structure of Matter

European Research Council

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VILLUM FONDEN through the QMATH Centre of Excellence (grant no. 10059)

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Annales Henri Poincaré

Volume

21

Issue

2

Pages

327 - 357

Publisher

Springer

Version

  • AM (Accepted Manuscript)

Rights holder

© Springer Nature

Publisher statement

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s00023-019-00855-7

Acceptance date

2019-09-30

Publication date

2019-10-10

Copyright date

2019

ISSN

1424-0637

eISSN

1424-0661

Language

  • en

Depositor

Dr Lukas Schimmer. Deposit date: 20 November 2023