On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach

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 $\det (\gamma -\Delta +V(x))$.

LaTeX, 8 pages, 7 eps figures, v2: new appendix, v3: discussions and ref added

References in corpus (2)

• Functionals of the Brownian motion, localization and metric graphs
• Quantum oscillations in mesoscopic rings and anomalous diffusion

Cited by in corpus (1)

• $\zeta$-regularised spectral determinants on metric graphs