Zero Modes of Quantum Graph Laplacians and an Index Theorem. / Bolte, Jens; Egger, Sebastian; Steiner, Frank.

In: Annales Henri Poincare, Vol. 16, No. 5, 05.2015, p. 1155-1189.

Research output: Contribution to journalArticle

Published

Documents

  • FKW_AP_new2

    Accepted author manuscript, 353 KB, PDF document

Links

Abstract

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 $\mf{S}$ 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.
Original languageEnglish
Pages (from-to)1155-1189
Number of pages35
JournalAnnales Henri Poincare
Volume16
Issue number5
Early online date11 Jul 2014
DOIs
Publication statusPublished - May 2015
This open access research output is licenced under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

ID: 20157037