Spectral determinants and zeta functions of Schr\"odinger operators on metric graphs

TL;DR

This paper derives the spectral determinant of Schrödinger operators on metric graphs with general vertex conditions, using spectral zeta functions and secular equations, advancing the understanding of spectral properties in quantum graph theory.

Contribution

It provides a general derivation of the spectral determinant for Schrödinger operators on metric graphs with broad vertex matching conditions, confirming recent conjectures.

Findings

Spectral zeta function derived using secular equations

Spectral determinant formulated for general vertex conditions

Results support recent conjectures in spectral graph theory

Abstract

A derivation of the spectral determinant of the Schr\"odinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of Kostrykin and Schrader. To formulate the spectral determinant we first derive the spectral zeta function of the Schr\"odinger operator using an appropriate secular equation. The result obtained for the spectral determinant is along the lines of the recent conjecture.