By N. N. Bogoliubov

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37) This symmetry in the potential induces a symmetry between the Jost functions analytic in the upper k-plane and the Jost functions analytic in the lower k-plane. 2 The inverse scattering transform for NLS 29 In turn, this symmetry of the Jost functions induces a symmetry in the scattering data. 3). 38a) ∗ . 39b) and consequently ρ(k) ¯ = ∓ρ ∗ (k) Im k = 0. 39a) it follows that k j is a zero of a(k) in the upper k-plane iff k ∗j is ¯ a zero for a(k) in the lower k-plane. 36), J = J¯ and k¯ j = k ∗j , c¯ j = ∓c∗j j = 1, .

54a) −2ikx ¯ (1) N (x, k)dk. 54b) Case of poles ¯ Suppose now that the potential is such that a(k) and a(k) have a ﬁnite number of simple zeros in the regions Im k > 0 and Im k < 0, respectively, which we J J¯ denote as k j , Im k j > 0 j=1 and k¯ j , Im k¯ j < 0 j=1 . We shall also assume ¯ ) = 0 for any ξ ∈ R. 31b). 56) with denoting the derivative with respect to the spectral parameter k. Note that the equations deﬁning the inverse problem for N (x, k) and N¯ (x, k) now depend J¯ J on the extra terms N j (x) j=1 and N¯ l (x) l=1 .

1 that if q, r ∈ L 1 (R), the Neumann series of the integral equations for M and N converge absolutely and uniformly (in x and k) in the upper k-plane, while the Neumann series of the integral equations for M¯ and N¯ converge absolutely and uniformly (in x and k) in the lower k-plane. These facts immediately imply that the Jost functions M(x, k) and N (x, k) are analytic func¯ tions of k for Im k > 0 and continuous for Im k ≥ 0, while M(x, k), and N¯ (x, k) are analytic functions of k for Im k < 0 and continuous for Im k ≤ 0.