The spectral transform in the semiclassical limit of a finite discrete NLS chain

The direct eigenvalue problem associated with an “inverse scattering” method for a finite nonlinear Schrödinger chain is studied in the semiclassical limit. In the case that the initial data for the chain are less than unity in modulus, the eigenvalue problem is unitary and the associated norming constants are real-valued. Formal asymptotic (WKB) analysis is performed, and formulas for the asymptotic spectral density and norming constants are obtained. They are supported by known facts about the discrete transform, rigorous asymptotics and numerical calculations.