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 $S_\mathcal{G}(γ)=\det(γ-Δ)$ 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 $\det(γ-Δ+V(x))$.

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