Annales Henri Poincaré

, Volume 16, Issue 5, pp 1155–1189 | Cite as

Zero Modes of Quantum Graph Laplacians and an Index Theorem

  • Jens Bolte
  • Sebastian EggerEmail author
  • Frank Steiner


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. To accommodate Laplacians with negative eigenvalues, it is necessary to define the Dirac operator on a suitable Kreĭn space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kreĭn-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.


Dirac Operator Zero Mode Trace Formula Hermitian Form Index Theorem 
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© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Royal HollowayUniversity of LondonEghamUK
  2. 2.Institut für Theoretische PhysikUniversität UlmUlmGermany
  3. 3.Observatoire de Lyon Centre de Recherche Astrophysique de Lyon, Université de Lyon CNRS UMR 5574: Université Lyon 1 and École Normale Supérieure de LyonSaint-Genis-LavalFrance

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