Spectral determinants and an Ambarzumian type theorem on graphs

TL;DR

This paper investigates the spectral properties of Schrödinger operators on equilateral graphs, deriving a spectral determinant, analyzing zero distributions, and establishing an Ambarzumian type theorem linking spectral data to potential functions.

Contribution

It introduces a method to compute the spectral determinant on graphs and proves a new inverse spectral result for zero potential identification.

Findings

Spectral determinant zeros follow polynomial roots.

One root equals the mean potential value.

Zero spectrum implies zero potential.

Abstract

We consider an inverse problem for Schr\"odinger operators on connected equilateral graphs with standard matching conditions. We calculate the spectral determinant and prove that the asymptotic distribution of a subset of its zeros can be described by the roots of a polynomial. We verify that one of the roots is equal to the mean value of the potential and apply it to prove an Ambarzumian type result, i.e., if a specific part of the spectrum is the same as in the case of zero potential, then the potential has to be zero.