Trace Formulae for quantum graphs with edge potentials

TL;DR

This paper derives trace formulas for quantum graphs with edge potentials, extending scattering methods to include potentials, and provides exact and semiclassical spectral formulas.

Contribution

It introduces an extension of the scattering approach to quantum graphs with potentials, deriving exact and semiclassical trace formulas.

Findings

Exact trace formulas for smooth and δ-potentials

Asymptotic semiclassical trace formula for smooth potentials

Extension of scattering approach to potential-including quantum graphs

Abstract

This work explores the spectra of quantum graphs where the Schr\"odinger operator on the edges is equipped with a potential. The scattering approach, which was originally introduced for the potential free case, is extended to this case and used to derive a secular function whose zeros coincide with the eigenvalue spectrum. Exact trace formulas for both smooth and $\delta$-potentials are derived, and an asymptotic semiclassical trace formula (for smooth potentials) is presented and discussed.