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

TL;DR

This paper derives a formula relating the spectral determinant of the Laplace operator on a metric graph formed by attaching two or more subgraphs at a vertex, extending to Schrödinger operators with potential.

Contribution

It provides a new explicit formula connecting the spectral determinants of combined graphs to those of individual subgraphs, generalizing previous results.

Findings

Derived a formula for spectral determinants of attached graphs

Extended the formula to Schrödinger operators with potential

Applicable to multiple graph attachments

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 Schr\"odinger operator $\det(\gamma-\Delta+V(x))$.