The Spectral Transform in the Semiclassical Limit of a Finite Discrete NLS Chain

The linear spectral problem associated with the inverse solution of a nite discrete nonlinear Schrr odinger chain is studied in the semiclassical limit. The discrete spectral problem is a recursion relation for a vector quantity, with boundary conditions, depending on initial data and a spectral parameter. WKB analysis is performed and then interpreted for the case that the quantities in the chain are less than one in modulus. In this case, the spectrum lies on the unit circle and an asymptotic density is obtained. The density is supported by known facts about the discrete spectra, numerical results, and rigorous results concerning the asymptotics of the solution of the spectral boundary-value problem. In addition, the norming constants in the spectral transform are positive in this special case, and a proposed asymptotic norming exponent is corroborated by numerical data.