Spectral analysis of Darboux transformations for the focusing NLS hierarchy

We study Darboux-type transformations associated with the focusing nonlinear Schrodinger 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) Dirac-type differential expression of the form [GRAPHICS] satisfying JM(q)J = M(q)*, where J is defined by J ((0)(1) (1)(0)) C, and C denotes the antilinear conjugation map in C-2, C(a, b)(T) = ((a) over bar, (b) over bar )T, a, b epsilon C. As one of our principal results, we prove that under the most general hypothesis q epsilon L-loc(1)(R) on q, the maximally defined operator D(q) generated by M(q) is actually J-self-adjoint in L-2(R)(2). Moreover, we establish the existence of Weyl-Titchmarsh-type solutions psi(+) (z,.) is an element of L-2 ([R, infinity))(2) and psi_ (z,.) is an element of L-2 ((-infinity, R]) for all R is an element of R of M(q)psi+/-(z) = zpsi+/-(z) for z in the resolvent set of D(q). The Darboux transformations considered in this paper are the analogue of the double commutation procedure familiar in the KdV and Schrodinger operator contexts. As in the corresponding case of Schrodinger operators, the Darboux transformations in question 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 is an element of L-loc(1)(R), associated with the Dirac-type operators D(q(N)) and D(q), respectively, amounts to the insertion of N complex conjugate pairs of L'(R)2-eigenvalues JZ1, Z1, - - - ZN, TN} into the spectrum o-(D(q)) of D(q), leaving the rest of the spectrum (especially, the essential spectrum sigma(D(q))) invariant, that is, sigma(D(q((N))) = sigma(D(q)) boolean OR {Z(1), (Z) over bar (1),..., Z(N), (Z) over bar (N)} sigma(e)(D(q((N))) = sigma(e)(D-(q)). 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-type transformations.