On a theorem of Ambarzumian (Q2764660)

The author considers the eigenvalue problem for the Dirac operator determined by system of differential equations NEWLINE\[NEWLINEu_2' +(V(x)+m)u_1 = \lambda u_1,\quad -u_1' +(V(x)-m)u_2= \lambda u_2,NEWLINE\]NEWLINE on the interval \([0,\pi]\) with the special boundary conditions \(u_1(0)=u_1(\pi)=0\). Under the assumption \(0<m\leq \frac{1}{2}\) (where the positivity of \(m\) is essential), the author proves the following uniqueness theorem (an analog of the Ambarzumian uniqueness result in the case of the one-dimensional Schödinder operator): the only potential \(V\) producing the spectrum identical to the spectrum corresponding to the case of the zero potential, \(V\equiv 0\), is the zero potential. The proof is based on a statement on generalized moments and allows a weakening of the conditions on the spectrum; particularly, one can assume that the spectrum coincides only asymptotically with those of the zero potential case.