Spectral Analysis of Darboux Transformations for the Focusing Nls Hierarchy

We study Darboux-type transformations associated with the focusing nonlinear Schrödinger equation (NLS−) and their effect on spectral properties of the underlying Lax operator. The latter is a formally J -selfadjoint (but non-self-adjoint) Dirac-type differential expression of the form M(q) = i d dx −q −q − d dx , 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 C2 , C(a, b) = (a, b), a, b ∈ C . As one of our principal results we prove that under the most general hypothesis q ∈ L1loc(R) on q, the maximally defined operator D(q) generated by M(q) is actually J -self-adjoint in L2(R)2. Moreover, we establish the existence of Weyl–Titchmarsh-type solutions Ψ+(z, ·) ∈ L2([R,∞))2 and Ψ−(z, ·) ∈ L2((−∞, R]) for all R ∈ R of M(q)Ψ±(z) = zΨ±(z) for z in the resolvent set of D. The Darboux transformations considered in this paper are the analog of the double commutation procedure familiar in the KdV and Schrödinger operator contexts. As in the corresponding case of Schrödinger operators, the Darboux transformations in question 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 ∈ L1loc(R), associated with the Dirac-type operators D q(N) and D(q), respectively, amounts to the insertion of N complex conjugate pairs of L2(R)2-eigenvalues {z1, z1, . . . , zN , zN} into the spectrum σ(D(q)) of D(q), leaving the rest of the spectrum (especially, the essential spectrum σe(D(q))) invariant, that is, σ D q = σ(D(q)) ∪ {z1, z1, . . . , zN , zN}, σe D q = σe(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. Date: October, 2002. 1991 Mathematics Subject Classification. Primary: 34L05, 34L40, 35Q51, 35Q55. Secondary: 34B20, 47A10.