Recovering the Shape of a Quantum Tree by Scattering Data

Abstract

We consider a scattering problem generated by the Sturm–Liouville equation on a tree which consists of an equilateral compact subtree and a half-infinite lead attached to its root. We assume that the potential on the lead is identically zero while the potentials on the finite edges are real. We show how to find the shape of the tree using the $S$-function of the scattering problem and the eigenvalues of the operators associated with the compact tree.