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