A solution to an Ambarzumyan problem on trees

TL;DR

This paper extends the Ambarzumyan theorem to Neumann Sturm-Liouville problems on trees with irrational edge length ratios, showing potential zero if spectra match zero potential case, using eigenvalue approximation and recursive formulas.

Contribution

It introduces a new Ambarzumyan theorem for Sturm-Liouville problems on trees with irrational edge ratios, expanding previous results.

Findings

Potential function must be zero if spectra match zero potential

Eigenvalues approximated using generalized pigeonhole argument

Recursive formulas for characteristic functions derived

Abstract

We consider the Neumann Sturm-Liouville problem defined on trees such that the ratios of lengths of edges are not necessarily rational. It is shown that the potential function of the Sturm-Liouville operator must be zero if the spectrum is equal to that for zero potential. This extends previous results and gives an Ambarzumyan theorem for the Neumann Sturm-Liouville problem on trees. To prove this, we compute approximated eigenvalues for zero potential by using a generalized pigeon hole argument, and make use of recursive formulas for characteristic functions.