Non-self-adjoint Zakharov–Shabat operator with a potential of the finite asymptotic values. I. Direct spectral and scattering problems

The Zakharov and Shabat equation for the scattering problem is studied: The estimates, analytical properties, and asymptotic expansions of the Jost solution are presented for a general class of the potentials Q(x) not vanishing at infinity. The existence of the similarity transformation is also shown. For Q(x) vanishing at infinity, the continuous part of the spectrum doubly degenerates. However, nonvanishing (finite) asymptotic values of Q(x) dissolve the degeneracy completely. The expansion theorem is given in C02(R) and for a class of Q(x) we prove that the Zakharov and Shabat equation yields a non-self-adjoint spectral operator in the Hilbert space in the sense of Dunford and Schwartz.