Spectral Analysis of Darboux Transformations for the Focusing NLS   Hierarchy
  We study Darboux transformations associated with the focusing nonlinear Schr\"odinger equation (NLS_-) and their effect on spectral properties of the underlying Lax operator. The latter is a formally J-self-adjoint (but non-self-adjoint) one-dimensional Dirac-type operator with potential q. As one of our principal results we prove that under the most general hypothesis of local integrability on q, the maximally defined operator D(q) generated by this Dirac operator is actually J-self-adjoint. We also establish the existence of Weyl--Titchmarsh-type solutions associated with D(q). The Darboux transformations considered in this paper guarantee that the resulting potentials q are locally nonsingular. Moreover, we prove that the construction of N-soliton NLS_- potentials q^{(N)} with respect to a general NLS_- background potential q, associated with Dirac operators D(q^{(N)}) and D(q), amounts to the insertion of N complex conjugate pairs of eigenvalues into the spectrum of D(q), leaving the rest of the spectrum (especially, the essential spectrum) invariant. These results are obtained by establishing the existence of bounded transformation operators which intertwine the background Dirac operator D(q) and the Dirac operator D(q^{(N)}) obtained after N Darboux transformations. 
Radu C. Cascaval, Fritz Gesztesy, Helge Holden, and Yuri Latushkin