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Krein's method in inverse scattering

A detailed discussion of the Krein's results (applicable for solving the inverse scattering problem) is given with complete proofs.


Introduction
The inverse scattering problem on half-axis consists of finding q(x) ∈ L 1,1 = q : q = q, , (1.5) and J ≥ 0 is the number of the negative eigenvalues. For simplicity we assume that there are no bound states. This assumption is removed in section 4.
This paper is a commentary to Krein's paper [1]. It contains not only a detailed proof of the results announced in [1] but also a proof of the new results not mentioned in [1]. In particular, it contains an analysis of the invertibility of the steps in the inversion procedure based on Krein's results, and a proof of the consistency of this procedure, that is, a proof of the fact that the reconstructed potential generates the scattering data from which it was reconstructed. Basic results are stated in Theorems 1.1 -1.4 below.
Note that and The function δ(k) is called the phase shift. One has S(k) = e 2iδ(k) .
We have changed the notations from [1] in order to show the physical meaning of the quantity (1.12): f (k) is the Jost function of the scattering theory. The function ψ(x, k)/f (k) is the solution to the scattering problem (see equation (2.39) below). Krein [1] calls the S-function, and we will show that (1.16) is the S-matrix used in physics. Assuming no bound states, one can solve the inverse scattering problem (ISP): A solution of the ISP based on the results of [1] consists of four steps: 1) Given S(k), find f (k) by solving the Riemann problem (2.37).

4) Define
and calculate the potential One can also calculate q(x) by the formula: There is an alternative (known) way, based on the Wiener-Levy theorem, to do step 1): Given S(k), find δ(k), the phase shift, then calculate the function and finally calculate The potential q ∈ L 1,1 generates the S-matrix S(k) with which we started provided that the following conditions (1.20) -(1.22) hold: (1.20) the overbar stands for complex conjugation, and (1.23) By the index (1.21) one means the increment of the argument of S(k) ( when k runs from −∞ to +∞ along the real axis) divided by 2π. The function (1.10) satisfies the equation Recall that we have assumed that there are no bound states.
In section 2 the above method is justified and the following theorems are proved: The diagram explaining the inversion method for solving ISP, based on Krein's results from [1], can be shown now: Indeed, if (1.20)-(1.22) hold, then a) and b) are contained in diagram (1.26), and c) follows from the known formula (e.g., [4], p.256) Then (still assuming (1.21)) one has: (1.28) Note that the general case of the inverse scattering problem on the half-axis, when can be reduced to the case ν = 0 by the procedure described in section 4 provided that S(k) is the S−matrix corresponding to a potential q ∈ L 1,1 (R + ). Necessary and sufficient conditions for such an S(k) are conditions (1.20)-(1.22) (see [4]).
Section 3 contains a discussion of the numerical aspects of the inversion procedure based on Krein's method. There are advantages in using this procedure (as compared with the Gelfand-Levitan procedure): integral equation (1.6), solving of which constitutes the basic step in the Krein inversion method, is a Fredholm convolution-type equation. Solving such an equation numerically leads to inversion of Toeplitz matrices, which can be done efficiently and with much less computer time than solving the Gelfand-Levitan equation (5.3). Combining Krein's and Marchenko's inversion methods yields the most efficient way to solve inverse scattering problems.
Indeed, for small x equation (1.6) can be solved by iterations since the norm of the integral operator in (1.6) is less than 1 for sufficiently small x, say 0 < x < x 0 . Thus q(x) can be calculated for 0 ≤ x ≤ x 0 2 by diagram (1.26). For x > 0 one can solve by iterations Marchenko's equation for the kernel A(x, y): where, if (1.21) holds, the known function F (x) is defined by the formula: Indeed, for x > 0 the norm of the operator in (1.29) is less than 1 ( [3]) and it tends to 0 as x → +∞. Finally let us discuss the following question: in the justification of both the Gelfand-Levitan and Marchenko methods, the eigenfunction expansion theorem and the Parseval relation play the fundamental role. In contrast, the Krein method apparently does not use the eigenfunction expansion theorem and the Parseval relation. However, implicitly, this method is also based on such relations. Namely, assumption (1.7) implies that the function S(k), that is, the S-matrix corresponding to the potential (1.19), has index 0. If, in addition, this potential is in by the formula generates the S-matrix S(k) (by formula (1.16)), and this S(k) satisfies conditions (1.20)-(1.22). Although the above conditions are verifiable in principle, they are not satisfactory because they are implicit, they are not formulated in terms of structural properties of the function H(t) (such as smoothness, rate of decay, etc.). In section 2 Theorems 1.1 -1.4 are proved. In section 3 numerical aspects of the inversion method based on Krein's results are discussed. In section 4 the ISP with bound states is discussed. In section 5 a relation between Krein's and Gelfand-Levitan's methods is explained.

Proofs
where the Parseval equality was used, Note that our argument shows that Before we prove Theorem 1.4, let us prove a simple lemma. For results of this type, see [2].
Lemma 2.1. The operator Proof. Let ||ϕ|| p := ||ϕ|| L p (R + ) . One has where we have used the assumption H(t) = H(−t). Similarly, Finally, using Parseval's equality, one gets: Since | H(k)| ≤ 2||H|| 1 one gets from (2.9) the estimate: To prove (2.6), one notes that Estimate ( For any fixed s > 0 one sees that sup x≥y ||h x (t, s)|| → 0 as y → ∞, where the norm here stands for any of the three norms L p (0, x), p = 1, 2, ∞. Therefore (2.12) and (2.11) imply since Γ(u, s) ∈ L 1 (R + ) for any fixed s > 0 and H(t) ∈ L 1 (R). Also where c j > 0 are some constants. Finally, by (2.6), one has; We will check below that and Using (2.17)-(2.18) again one gets The first condition is obvious (in [1] there is a misprint: it is written that ψ(0, k) = 1), and the second condition follows from (1.10) and (2.15): Let f (x, k) be the Jost solution to (1.24) which is uniquely defined by the asymptotics Since f (x, k) and f (x, −k) are lineraly independent, one has where c 1 , c 2 are some constants independent of x but depending on k. From (2.31) and (2.33) one gets Indeed, the choice of c 1 and c 2 guarantees that the first condition (2.31) is obviously satisfied, while the second follows from the Wronskian formula: Let us start with a discussion of the inversion steps 1) -4) described in the introduction. Then we discuss the uniqueness of the solution to ISP and the consistency of the inversion method, that is, the fact that q(x), reconstructed from S(k) by steps 1) -4), generates the original S(k).
Let us go through steps 1) -4) of the reconstruction method and prove their invertibility. The consistency of the inversion method follows from the invertibility of the steps of the inversion method. where (see [3]) so that f ′ (k) is bounded for all k ∈ R, as claimed. Note that The converse step f (k) ⇒ S(k) is trivial: If ind R S = 0 then f (k) is analytic in C + , f (k) = 0 in C + , f (k) = 1 + O 1 k as |k| → ∞, k ∈ C + , and (2.48) holds.
The converse step is also done by formula (1.17): Fourier inversion gives |f (k)| 2 = f (k)f (−k), and factorization yields the unique f (k), since f (k) does not vanish in C + and tends to 1 at infinity.
Step Note that the step q(x) ⇒ f (k) can be done by solving the uniquely solvable integral equation (see [4]): with q ∈ L 1,1 (R + ), and then calculating f (k) = f (0, k).
It is not proved in [1] that q(x) defined in (1.19) (and obtained as a final result of steps 1) -4)) generates the scattering matrix S(k) with which we started the inversion.
Let us now prove this. We have already discussed the following diagram: To close this diagram and therefore establish the basic one-to-one correspondence This is done by the scheme (2.54). Note that the step q(x) ⇒ A(x) requires solving Riccati equation (1.19) with the boundary condition A(0) = 2H(0). Existence of the solution to this problem on all of R + is guaranteed by the assumptions (1.20)-(1.22). The fact that these assumptions imply q(x) ∈ L 1,1 (R + ) is proved in [3] and [4]. Theorem 1.1 is proved. ✷ Uniqueness theorems for the inverse scattering problem are not given in [1], [2]. They can be found in [3]- [7]. 3 Numerical aspects of the Krein inversion procedure.
The main step in this procedure from the numerical viewpoint is to solve equation (1.6) for all x > 0 and all 0 < s < x, which are the parameters in equation (1.6).
Since equation (1.6) is an equation with the convolution kernel, its numerical solution involves inversion of a Toeplitz matrix, which is a well developed area of numerical analysis. Moreover, such an inversion requires much less computer memory and time than the inversion based on the Gelfand-Levitan or Marchenko methods. This is the main advantage of Krein's inversion method.
This method may become even more attractive if it is combined with the Marchenko method. In the Marchenko method the equation to be solved is ( [3], [4]): where F (x) is defined in (1.23) and is known if S(k) is known, the kernel A(x, y) is to be found from (3.1) and if A(x, y) is found then the potential is recovered by the formula: Equation (3.1) can be written in operator form: The operator F x is a contraction mapping in the Banach space L 1 (x, ∞) for x > 0. The operator H x in (1.6) is a contraction mapping in L ∞ (0, x) for 0 < x < x 0 , where x 0 is chosen to that Therefore it seems reasonable from the numerical point of view to use the following approach: and W (k) = k k + iγ w(k) if ind R S(k) = −2J − 1, (4.2) where γ > 0 is chosen so that γ = k j , 1 ≤ j ≤ J. Then one defines Since ind R w 2 (k) = 2J and ind R W 2 (k) = 2J + 1, one has ind R S 1 (k) = 0. (4.5) The theory of section 2 applies to S 1 (k) and yields q 1 (x). From q 1 (x) one gets q(x) by adding bound states −k 2 j and norming constants s j using the known procedure (e.g. see [3]).

Relation between Krein's and GL's methods.
The GL (Gelfand-Levitan) method consists of the following steps (see [4], for example): Step Step 2. Solve the integral equation for K(x, y): K(x, y) + L(x, y) + Krein's function (see (1.17)) can be written as follows: |f (k)| −2 − 1 cos(kt)dk. (5.5) Thus the relation between the two methods is given by the known formula: In fact, the GL method deals with the inversion of the spectral foundation dρ of the operator − d 2 dx 2 + q(x) defined in L 2 (R + ) by the Dirichlet boundary condition at x = 0. However, if ind R S(k) = 0 (no bound states and no resonance at k = 0), then (see e.g. [4]): so dρ in this case is uniquely defined by f (k).