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 journalArticlepeer-review

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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 journalArticlepeer-review

Harvard

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

APA

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

Vancouver

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

Author

Bolte, Jens ; Egger, Sebastian ; Steiner, Frank. / Zero Modes of Quantum Graph Laplacians and an Index Theorem. In: Annales Henri Poincare. 2015 ; Vol. 16, No. 5. pp. 1155-1189.

BibTeX

@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",

}

RIS

TY - JOUR

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

AU - Bolte, Jens

AU - Egger, Sebastian

AU - Steiner, Frank

PY - 2015/5

Y1 - 2015/5

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.

U2 - 10.1007/s00023-014-0347-z

DO - 10.1007/s00023-014-0347-z

M3 - Article

VL - 16

SP - 1155

EP - 1189

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 5

ER -