On the spectrum of the Laplace operator of metric graphs attached at a vertex

doi:10.1088/1751-8113/41/8/085207

On the spectrum of the Laplace operator of metric graphs attached at a vertex—spectral determinant approach

Christophe Texier 2008 J. Phys. A: Math. Theor. 41 085207 (9pp) doi: 10.1088/1751-8113/41/8/085207

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Christophe Texier
Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626 du CNRS, Université Paris-Sud, Bât. 100, F-91405 Orsay Cedex, France
and
Laboratoire de Physique des Solides, UMR 8502 du CNRS, Université Paris-Sud, Bât. 510, F-91405 Orsay Cedex, France

Abstract. We consider a metric graph $\mathcal{G}$ made of two graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ attached at one point. We derive a formula relating the spectral determinant of the Laplace operator $S_\mathcal{G}(\gamma)=\det(\gamma-\Delta)$ in terms of the spectral determinants of the two subgraphs. The result is generalized to describe the attachment of n graphs. The formulae are also valid for the spectral determinant of the Schrödinger operator $\det(\gamma-\Delta+V(x))$ .

PACS numbers: 02.70.Hm, 02.10.Ox

Print publication: Issue 8 (29 February 2008)
Received 1 June 2007, in final form 20 December 2007
Published 12 February 2008

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On the spectrum of the Laplace operator of metric graphs attached at a vertex—spectral determinant approach

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