On the stability in Ambarzumian theorems

We provide extensions of the classical Ambarzumian theorem for bounded C 3 domains of any dimension. The simple proof is based on classical spectral function asymptotics. We prove a stability property by showing that if the perturbation of the eigenvalues of the zero potential is small in some sense then the L 2-norm of the potential is also small. The problem is motivated by connections to a number of applications. (paper)