An Ambarzumian type theorem on graphs with odd cycles

TL;DR

This paper proves an Ambarzumian type theorem for Schrödinger operators on certain graphs, showing that spectral data determines zero potential when specific spectral conditions are met on graphs with odd cycles.

Contribution

It establishes a new inverse spectral result for Schrödinger operators on graphs with odd cycles, extending classical Ambarzumian theorems to this setting.

Findings

Spectral data uniquely determines zero potential under given conditions.

The theorem applies to graphs with at least two odd cycles sharing a vertex.

The result generalizes classical inverse spectral theorems to graph structures.

Abstract

We consider an inverse problem for Schr\"odinger operators on a connected equilateral graph $G$ with standard matching conditions. The graph $G$ consists of at least two odd cycles glued together at a common vertex. We 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.