$\zeta$-regularised spectral determinants on metric graphs

TL;DR

This paper reviews spectral determinants of Schrödinger operators on metric graphs, introduces a simple derivation for their $;zeta$-regularised form, and explores boundary conditions and connections to combinatorial graph formulas.

Contribution

It provides a new simple derivation of the $;zeta$-regularised spectral determinant using the Roth trace formula and discusses general boundary conditions.

Findings

Derivation of $;zeta$-regularised spectral determinant

Analysis of boundary conditions at vertices

Discussion of relations to combinatorial graph formulas

Abstract

Several general results for the spectral determinant of the Schr\"odinger operator on metric graphs are reviewed. Then, a simple derivation for the $\zeta$-regularised spectral determinant is proposed, based on the Roth trace formula. Two types of boundary conditions are studied: functions continuous at the vertices and functions whose derivative is continuous at the vertices. The $\zeta$-regularised spectral determinant of the Schr\"odinger operator acting on functions with the most general boundary conditions is conjectured in conclusion. The relation to the Ihara, Bass and Bartholdi formulae obtained for combinatorial graphs is also discussed.