Ambarzumian's theorem for trees

Abstract

The classical Ambarzumian's Theorem for Schrödinger operators -D2 + q on an interval, with Neumann conditions at the endpoints, says that if the spectrum of (-D2 + q) is the same as the spectrum of (-D2) then q = 0. This theorem is generalized to Schrödinger operators on metric trees with Neumann conditions at the boundary vertices.