On a theorem of Ambarzumian

We consider the eigenvalue problem for the one-dimensional (stationary) Dir ac operator with some boundary conditions. We prove that if the spectrum is the same as the spectrum belonging to the zero potential, then the potenti al is actually zero. The analogous statement for the Schrodinger operator i s due to Ambarzumian. The proof is based on the fact that the (generalized) moments of a function cannot have alternating signs unless the moments are zero (see 2).