Ambarzumyan’s Theorem for the Dirac Operator on Equilateral Tree Graphs

Abstract

The classical Ambarzumyan’s theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator \( - {{{d^2}} \over {d{x^2}}} + q\) with an integrable real-valued potential q on [0, π] are {n²: n ≥ 0}, then q = 0 for almost all x ∈ [0, π]. In this work, the classical Ambarzumyan’s theorem is extended to the Dirac operator on equilateral tree graphs. We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case, then the potential is identically zero.