Generalised Dirac-Schrödinger operators and the Callias Theorem
We consider generalised Dirac-Schrödinger operators, consisting of a self-adjoint elliptic first-order differential operator $\mathcal {D}$ with a skew-adjoint ‘potential’ given by a (suitable) family of unbounded operators. The index of such an operator represents the pairing (Kasparov produ...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001579/type/journal_article |
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| author | Koen van den Dungen |
| author_facet | Koen van den Dungen |
| author_sort | Koen van den Dungen |
| collection | DOAJ |
| description | We consider generalised Dirac-Schrödinger operators, consisting of a self-adjoint elliptic first-order differential operator
$\mathcal {D}$
with a skew-adjoint ‘potential’ given by a (suitable) family of unbounded operators. The index of such an operator represents the pairing (Kasparov product) of the K-theory class of the potential with the K-homology class of
$\mathcal {D}$
. Our main result in this paper is a generalisation of the Callias Theorem: the index of the Dirac-Schrödinger operator can be computed on a suitable compact hypersurface. Our theorem simultaneously generalises (and is inspired by) the well-known result that the spectral flow of a path of relatively compact perturbations depends only on the endpoints. |
| format | Article |
| id | doaj-art-04fd3c6bb802498abccd3d3c777cd876 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-04fd3c6bb802498abccd3d3c777cd8762025-01-24T05:20:13ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.157Generalised Dirac-Schrödinger operators and the Callias TheoremKoen van den Dungen0https://orcid.org/0000-0002-0252-9579Mathematisches Institut, Universität Bonn, Endenicher Allee 60, D-53115, GermanyWe consider generalised Dirac-Schrödinger operators, consisting of a self-adjoint elliptic first-order differential operator $\mathcal {D}$ with a skew-adjoint ‘potential’ given by a (suitable) family of unbounded operators. The index of such an operator represents the pairing (Kasparov product) of the K-theory class of the potential with the K-homology class of $\mathcal {D}$ . Our main result in this paper is a generalisation of the Callias Theorem: the index of the Dirac-Schrödinger operator can be computed on a suitable compact hypersurface. Our theorem simultaneously generalises (and is inspired by) the well-known result that the spectral flow of a path of relatively compact perturbations depends only on the endpoints.https://www.cambridge.org/core/product/identifier/S2050509424001579/type/journal_article19K5619K3558J20 |
| spellingShingle | Koen van den Dungen Generalised Dirac-Schrödinger operators and the Callias Theorem Forum of Mathematics, Sigma 19K56 19K35 58J20 |
| title | Generalised Dirac-Schrödinger operators and the Callias Theorem |
| title_full | Generalised Dirac-Schrödinger operators and the Callias Theorem |
| title_fullStr | Generalised Dirac-Schrödinger operators and the Callias Theorem |
| title_full_unstemmed | Generalised Dirac-Schrödinger operators and the Callias Theorem |
| title_short | Generalised Dirac-Schrödinger operators and the Callias Theorem |
| title_sort | generalised dirac schrodinger operators and the callias theorem |
| topic | 19K56 19K35 58J20 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001579/type/journal_article |
| work_keys_str_mv | AT koenvandendungen generaliseddiracschrodingeroperatorsandthecalliastheorem |