Eigenvalue analysis of the lax operator for the one-dimensional cubic nonlinear defocusing schrodinger equation

We characterize the location and number of eigenvalues for the Lax operator associated to the one-dimensional cubic nonlinear defocusing Schro"\dinger equation. With the help of a newly discovered unitary matrix, the analysis reduces to the study of the spectral problem for a unitarily equivalent operator, which involves only the amplitude and the phase velocity of the potential. Examples of potentials with special amplitude and phase velocity are investigated.