Inverse Problem for the \(L\)-Operator in the Lax Pair of the Boussinesq Equation on the Circle

We consider a third-order non-self-adjoint operator which is an \(L\) -operator in the Lax pair for the Boussinesq equation on the circle. We construct a mapping from the set of operator coefficients to the set of spectral data, similar to the corresponding mapping for the Hill operator constructed by E. Korotyaev. We prove that, in a neighborhood of zero, our mapping is analytic and one-to-one.