A solution to an Ambarzumyan problem on trees

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.

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