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The inverse spectral theory for a self-adjoint one-dimensional Dirac operatorassociated periodic potentials is formulated via a Riemann-Hilbert problemapproach. The resulting formalism is also used to solve the initial valueproblem for the nonlinear Schrodinger (NLS) equation. A uniqueness theorem forthe solutions of the Riemann-Hilbert problem is established, which provides anew method for obtaining the potential from the spectral data. Two additional,scalar Riemann-Hilbert problems are also formulated that provide conditions forthe periodicity in space and time of the solution generated by arbitrary setsof spectral data. The formalism applies for both finite-genus andinfinite-genus potentials. The formalism also shows that only a single set ofDirichlet eigenvalues is needed in order to uniquely reconstruct the potentialof the Dirac operator and the corresponding solution of the defocusing NLSequation, in contrast with the representation of the solution of the NLSequation via the finite-genus formalism, in which two different sets ofDirichlet eigenvalues are used.

Spectral theory for self-adjoint Dirac operators with periodic  potentials and inverse scattering transform for the defocusing nonlinear  Schroedinger equation with periodic boundary conditions