Wave operators for the matrix Zakharov–Shabat system

In this article, we prove the similarity and, in the focusing case, the J-unitary equivalence of the free Hamiltonian and the restriction of the full Hamiltonian to the maximal invariant subspace on which its spectrum is real for the matrix Zakharov–Shabat system under suitable conditions on the potentials. This restriction of the full Hamiltonian is shown to be a scalar-type spectral operator whose resolution of the identity is evaluated. In the focusing case, the restricted full Hamiltonian is an absolutely continuous, J-self-adjoint non-J-definitizable, operator allowing a spectral theorem without singular critical points. To illustrate the results, two examples are provided. © 2010 American Institute of Physics. doi:10.1063/1.3377048