Eigenvalue analysis of the Lax operator for the one-dimensional cubic nonlinear defocusing Schrödinger equation

We analyze the eigenvalues of the Lax operator associated to the one-dimensional cubic nonlinear defocusing Schrödinger equation. With the help of a newly discovered unitary matrix, it reduces to the study of a unitarily equivalent operator, which involves only the amplitude and the phase velocity of the potential. For a specific kind of potentials which satisfy nonzero boundary conditions, the eigenvalues of the Lax operator are characterized via a family of compact operators.