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

Research output: Contribution to journal › Article › peer-review

Published

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

Research output: Contribution to journal › Article › peer-review

Bolte, J, Egger, S & Steiner, F 2015, 'Zero Modes of Quantum Graph Laplacians and an Index Theorem', *Annales Henri Poincare*, vol. 16, no. 5, pp. 1155-1189. https://doi.org/10.1007/s00023-014-0347-z

Bolte, J., Egger, S., & Steiner, F. (2015). Zero Modes of Quantum Graph Laplacians and an Index Theorem. *Annales Henri Poincare*, *16*(5), 1155-1189. https://doi.org/10.1007/s00023-014-0347-z

Bolte J, Egger S, Steiner F. Zero Modes of Quantum Graph Laplacians and an Index Theorem. Annales Henri Poincare. 2015 May;16(5):1155-1189. https://doi.org/10.1007/s00023-014-0347-z

@article{069d58dd9b2e4c268e0d861cdd913c42,

title = "Zero Modes of Quantum Graph Laplacians and an Index Theorem",

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 intomomentum-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. ",

author = "Jens Bolte and Sebastian Egger and Frank Steiner",

year = "2015",

month = may,

doi = "10.1007/s00023-014-0347-z",

language = "English",

volume = "16",

pages = "1155--1189",

journal = "Annales Henri Poincare",

issn = "1424-0637",

publisher = "Birkhauser Verlag Basel",

number = "5",

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T1 - Zero Modes of Quantum Graph Laplacians and an Index Theorem

AU - Bolte, Jens

AU - Egger, Sebastian

AU - Steiner, Frank

PY - 2015/5

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N2 - 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 intomomentum-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.

AB - 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 intomomentum-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.

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DO - 10.1007/s00023-014-0347-z

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VL - 16

SP - 1155

EP - 1189

JO - Annales Henri Poincare

JF - Annales Henri Poincare

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