### A Solution to an Ambarzumyan Problem on Trees

**Law, C-K ***(National Sun Yat-sen)*

Thursday 29 July 2010, 11:15-12.00

Seminar Room 1, Newton Institute

#### Abstract

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.

#### Presentation

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