A Solution to an Ambarzumyan Problem on Trees

The classical Ambarzumyan problem states that when the eigenvalues $\lambda_n$ of a Neumann Sturm-Liouville operator defined on $[0,\pi]$ are exactly $n^2$, then the potential function $q=0$. In 2007, Carlson and Pivovarchik showed the Ambarzumyan problem for the Neumann Sturm-Liouville operator defined on trees where the edges are in rational ratio. We shall extend their result to show that for a general tree, if the spectrum $\sigma(q)=\sigma(0)$, then $q=0$. In our proof, we develop a recursive formula for characteristic functions, together with a pigeon hole argument. This is a joint work with Eiji Yanagida of Tokyo Institute of Technology.