 02445 Radu C. Cascaval, Fritz Gesztesy, Helge Holden, and Yuri Latushkin
 Spectral Analysis of Darboux Transformations for the Focusing NLS Hierarchy
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Oct 31, 02

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Abstract. 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
Jselfadjoint (but nonselfadjoint) onedimensional Diractype 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 Jselfadjoint. We also establish the existence of WeylTitchmarshtype 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 Nsoliton 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.
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