Wave operators for the matrix Zakharov-Shabat system
van der Mee, C.
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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. (C) 2010 American Institute of Physics. [doi:10.1063/1.3377048]