DIRECT SCATTERING OF AKNS SYSTEMS WITH L POTENTIALS

In this article the Jost solutions of the AKNS system with suitably weighted L2 potential are constructed as Hardy space perturbations of their space-infinity asymptotics. The reflection coefficients are proven to be L2-functions when the transmission coefficients are L∞-functions.

1. Introduction.In this article we discuss direct scattering for the AKNS system [16,1,12,6,2,3,13] iJ ∂X ∂x (x, λ) − iQ(x)X(x, λ) = λX(x, λ), where the potentials q(x) and r(x) have their entries in L 2 (R), λ is a spectral parameter, and I p denotes the p × p unit matrix.In the defocusing case (σ = −1) and the focusing case (σ = 1) we have the symmetry relations r(x) = σq(x) † and hence Q(x) † = σQ(x), where the dagger denotes complex conjugate matrix transposition.Contrary to the usual situation in the literature, the potentials are not assumed to be L 1 (as in [2,3,13]) or to belong to the Schwartz class (as in [6]) but to satisfy The main application of the direct scattering theory of the Schrödinger equation on the line and the AKNS system is to solve the Cauchy problem of certain integrable nonlinear evolution equations by means of the inverse scattering transform (IST) method.This means that the time evolution of the potential is transformed, by means of the IST, into the elementary time evolution of the scattering data.This has led to an algorithm to solve the Korteweg-de Vries (KdV) equation by using the scattering theory of the Schrödinger equation on the line and an algorithm to solve the nonlinear matrix Schrödinger (NLS) equation by using that of the AKNS system.A natural question to answer is how to define a sufficiently extensive class of potentials and a sufficiently extensive class of scattering data such that there is a 1, 1-correspondence between potentials and scattering data by means of the IST (the so-called characterization problem).
Characterization of scattering data of linear differential systems similar to (1.1) has a long history.Melin [11] has characterized the scattering data of the Schrödinger equation on the line with real potentials Q(x) such that (1 + |x|)Q(x) is integrable.Previous results for real potentials Q(x) such that (1 + x 2 )Q(x) is integrable, are due to Marchenko [10].

CORNELIS VAN DER MEE
For the AKNS system with L 1 potentials, Demontis and Van der Mee [5] have given a 1, 1correspondence between L 1 -potentials without spectral singularities and suitable scattering data.We refer to [15,3,4] for prior partial results.Unfortunately, these characterization results are not invariant under time evolution.In fact, time evolution according to the matrix NLS equation might lead to potentials and scattering data not belonging to the two classes in 1, 1-correspondence.Moreover, one cannot formulate even the simplest of the infinitely many conservation laws for every time dependent potential belonging to the class.
This article is meant as a contribution towards a characterization result, where a large class of potentials and a large class of scattering data are put in such a 1, 1-correspondence that either class is invariant under time evolution according to the matrix nonlinear Schrödinger (NLS) equation.Van der Mee [14] has given the following partial solution to the time-evolution invariant characterization problem: a) Assuming a reflection coefficient to be continuous in λ ∈ R, vanishing as λ → ±∞, and L 2 , plus reasonable bound state data, a unique L 2 potential can be constructed; b) a dense linear subspace of potentials with entries in L 1 (R) ∩ L 2 (R) is required to arrive at scattering data within the designated, time evolution invariant, class.
In this article we focuss on particular details of the construction: How to define Jost solutions and scattering coefficients for certain non-L 1 -potentials, and how to prove the reflection coefficients to be L 2 .
The AKNS operator iJ d dx I m+n − iQ can be defined in a natural way on the orthogonal sum of m + n copies of L 2 (R) for L 1 loc potentials [8].In the defocusing case the AKNS operator has a unique selfadjoint extension.For L 2 potentials this operator has the same domain as the free AKNS operator i d dx J (namely, the direct sum of m + n copies of the first Sobolev space) and has the real line as its essential spectrum [9].
Let us discuss the contents of the various sections.In Sec. 2 we write the Jost solutions in triangular representation form and iterate the resulting integral equations for the kernel functions in the L 2 norm.Under condition (1.3), this will lead to Jost solutions which are still analytic in the spectral variable in the upper or lower half-plane but are no longer continuous in the spectral variable when approaching the real line.Instead the Jost solutions will belong to suitable Hardy spaces of analytic functions.In Sec. 3 we construct the reflection and transmission coefficients as L 1 loc functions of λ ∈ R. Assuming the absence of spectral singularities, the reflection coefficients are shown to have their entries in L 2 (R).
2. Jost solutions.The Jost matrices Ψ(λ, x) and Φ(λ, x) are those solutions to (1.1) which behave as e −iλxJ [I m+n + o(1)] as x → +∞ and x → −∞, respectively.They can be partitioned into Jost solutions as follows where ψ(λ, x) and φ(λ, x) are (m + n) × m and ψ(λ, x) and φ(λ, x) are (m + n) × n matrices.For L 1 potentials and λ ∈ R their existence can be proved in the traditional way [2,3,13] by iterating Volterra integral equations.Here we prove their existence, for almost every λ ∈ R, for potentials satisfying (1.3) by a different technique.For L 1 loc -potentials, Klaus [8] has constructed Jost solutions by using arbitrary bases of the linear spaces of AKNS solutions that are L 2 on either the left or the right half-line.In this article we do not pursue his construction but follow a more direct route to Jost solutions instead.
Writing the triangular representations where the kernel functions K(x, y) and M(x, y) can be decomposed as we obtain the integral equations as well as (2.3d) For potentials with L 1 entries, (2.2) and (2.3) are easily shown to be uniquely solvable by iteration [2,13], yielding We now establish the unique solvability of the integral equations (2.2) and (2.3) for potentials Q(x) satisfying (1.3).
Proof.We only prove the first statement.Estimating (2.2a) we get

CORNELIS VAN DER MEE
Iterating (2.2b) we obtain Schematically, these two inequalities can be written as (2.4a) (2.4b) Taking the square root and applying the triangle inequality we obtain , . With some effort we get the estimate Using Gronwall's inequality we get , where we have used that Then the above inequalities (2.4) imply that for so that the left-hand sides are finite if B(x 1 ) < 2 −1/2 .Taking points , we can apply the same argument on the successive intervals [ξ s−1 , ξ s ].In fact, from (2.4) we get which proves the finiteness of The proofs for the other kernel functions are identical.
Theorem 2.1 implies that, under the condition where x ∈ R.
We now introduce the Hardy spaces H 2 (C ± ) as the complex Hilbert spaces of those analytic functions f (λ) in λ ∈ C ± such that is finite.Then H 2 (C ± ) coincides with the image of the L 2 functions supported on R ± under Fourier transformation [7].Using JQ(x)J = −Q(x) and assuming X −1 exists, Eq. (1.1) implies that where vanishes for any solution X to (1.1) and Y to (2.5).Taking Y = Ψ −1 or Y = Φ −1 , we can derive the triangular representations where the kernel functions K(y, x) and M(y, x) can be decomposed as .
For later use, we proceed as above and derive the analogs of (2.2) and (2.3) as well as the following results.
for x ≥ x 0 .Then for x ≥ x 0 there exists a unique kernel function K(y, x) such that (2.7a) is satisfied and as L 1 loc functions of λ ∈ R which do not depend on x ∈ R. It is then easily verified that a l1 (λ) and a r4 (λ) extend to functions that are analytic in λ ∈ C + , whereas a r1 (λ) and a l4 (λ) extend to functions that are analytic in λ ∈ C − .In fact, Ψ(x, λ) and Φ(x, λ) are both square matrix solutions of the same first order system (1.1) and hence one is obtained from the other by postmultiplication by a square matrix a r/l (λ) not depending on x ∈ R.
In this paper we make the following no-spectral-singularity assumption: 1 a l1 (λ) (or a r4 (λ) ) and a r1 (λ) (or a l4 (λ) ) are both almost everywhere positive and their reciprocals are essentially bounded in λ ∈ R.