| Autor: |
Bolte, Jens, Egger, Sebastian, Steiner, Frank |
| Rok vydání: |
2013 |
| Předmět: |
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| Zdroj: |
Annales Henri Poincar\'e, Volume 16, Issue 5, pp 1155-1189, 2015 |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.1007/s00023-014-0347-z |
| Popis: |
We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper the number of zero modes is expressed in terms of the trace of a unitary matrix $\mathfrak{S}$ that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part a Dirac operator is defined whose square is related to the Laplacian. In order to accommodate Laplacians with negative eigenvalues it is necessary to define the Dirac operator on a suitable Kre\u{\i}n space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kre\u{\i}n-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator. |
| Databáze: |
arXiv |
| Externí odkaz: |
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