SPECTRAL SHIFT FUNCTIONS IN THE FIXED ENERGY INVERSE SCATTERING

In this paper the notion of the Krein spectral shift function is extended to the radial Schrödinger operator with fixed energy. Then we analyze the connections between the tail of the potential and the decay rate of the fixedenergy phase shifts. Finally we extend former results on the uniqueness of the fixed-energy inverse scattering problem to a general class of potentials.


1.
Introduction. The notion of the spectral shift function, introduced by M. G. Krein has become an important tool in the inverse spectral theory of Schrödinger and other operators. The interested reader can consult the review paper of Birman and Yafaev [6] and many other publications, e.g. Simon [28], Gesztesy and Simon [12], [13], Gesztesy and Holden [10], Gesztesy and Makarov [11] and so on. In the present paper we consider analogous notions for the three-dimensional inverse potential scattering with fixed energy in case of spherically symmetrical potentials. This is described by the radial Schrödinger operator (τ f )(r) = −(r 2 f (r)) − 1 4 f (r) + r 2 (q(r) − 1)f (r).
The quantity δ(ν) is the phase shift corresponding to the spectral parameter ν. It is defined by (4) only modulo π but it can be defined as an analytic function of Remark 1. Often the equation hy = − ν 2 r 2 y, hy = −y" + (q(r) − 1)y − 1 4r 2 y is called radial Schrödinger equation, see e.g. [7], (1.1.6) or [20], Ch. 20.4. It is easy to check that h = 1 r τ 1 r . The operator τ used e.g. in [18] has the technical advantage that it defines selfadjoint operators in the unweighted L 2 space. The solutions of τ f = −ν 2 f are of order r ν−1/2 or r −ν−1/2 near zero, so the subspace of L 2 -solutions near r = 0 is one-dimensional if ν > 0 and ν = 0. On the other hand all solutions are asymptotically cos(r + α)/r for large r with some constants α thus all solutions belong to L 2 at infinity. So, by the classical Weyl terminology τ is in the limit point case at zero and in the limit circle case at infinity, the defect indices are (1,1) and the selfadjoint extensions are given by introducing a boundary condition at infinity. That is, if θ ∈ [0, π) is a fixed parameter, τ is selfadjoint with the domain where W R (f, g) = r 2 (f g − f g) is the weighted (or radial) Wronskian, see e.g. [18]. If ϕ ∈ D θ then λ = −ν 2 is an eigenvalue of τ . It is known that the eigenvalues λ n (θ) are negative and tend to −∞ by the rate The phase shifts are related to the scattering amplitude by the well-known formula where P n (t) are the Legendre polynomials. Thus the physical phase shifts δ n = δ(n + 1/2) are of primary interest in the applications. However Regge [26] proposed the investigation of δ(ν) with complex ν as early as in 1959. The Gelfand-Levitan inverse spectral theory has been extended to this complex setting e.g. in Loeffel [18] and Burdet, Giffon and Predazzi [9]. The idea of taking complex phase shifts proved to be fruitful in investigating inverse scattering problems, see e.g. [18]. In the present paper two uniqueness results are based on the Regge uniqueness theorem, using δ(ν), ν > 0; details are given later. Concerning the notion of the spectral shift function belonging to the operators τ the following statements will be proved. In analogy with Gesztesy and Simon [12] we prove Theorem 1.1. Let 0 ≤ θ 1 < θ 2 < π and suppose (2). If the domain (5) of the operator τ j is defined by the parameter θ j , j = 1, 2, then there exists a measurable function 0 ≤ ξ θ1,θ2 (t) ≤ 1 such that The range of (τ 2 + ν 2 ) −1 − (τ 1 + ν 2 ) −1 consists of the constant multiples of ϕ(., ν). Under the additional condition the spectral shift function ξ θ1,θ2 (t) uniquely determines θ 1 , θ 2 and the potential q(r) a.e. in [0, ∞).
Now let θ be fixed, τ be the operator with domain (5) and for a constant r 0 > 0 let τ r0 be the selfadjoint operator with domain Consider the function consists of functions parallel with ϕ on (0, r 0 ), with ψ on (r 0 , ∞) and continuous at r 0 . The function F (z) is Herglotz (i.e. z > 0 implies F (z) > 0) and there exists a measurable function 0 ≤ ξ r0 (t) ≤ 1 called Krein spectral shift function such that In case of Schrödinger operators there are explicit formulae (called trace formulae) expressing special values of the potential by the Krein function, see e.g. Gesztesy and Simon [13], Gesztesy and Holden [10] and Rybkin [27]. The counterpart of these formulae in the situation of Theorem 1.2 is Theorem 1.3. Suppose (2). If q is right and left Lebesgue continuous at r 0 then In case of Theorem 1.1 we can state the vague claim that the characteristics of the behavior of the potential at infinity can be expressed from the Krein function. Since the phase shifts can be expressed by the Krein function, we formulate the results directly for the phase shifts δ(ν). Below we obtain an explicit asymptotics for the phase shifts with an estimation of the remainder. Theorem 1.4. Suppose that ν > 1/3 is sufficiently large: Then For large ν the Bessel function J ν is extremely small in fixed finite intervals, so (12) implies that the asymptotical properties of δ(ν) are mostly influenced by the tail of the potential. Some special cases are listed below, where the tail of q has polynomial, exponential or superexponential decay and where q has compact support. In all cases the asymptotics of the potential can be reconstructed from the asymptotics of phase shifts of large indices; in the last case the support [0, a] and q(a − 0) can be recovered.

Corollary 1. Under the condition (2)
• If q(r) = cr −s (1 + o(1)) as r → ∞ with some constants c = 0 and s > 2 then for ν → ∞ • If q(r) = c r e −ar (1 + o (1)) with c = 0 and a > 0 then • If q(r) = ce −a 2 r 2 (1 + o (1)) with c = 0 and a > 0 then • If q = 0 for r > a and q is left Lebesgue continuous in a in the sense that Our last topic is the uniqueness of the inverse scattering problem with fixed energy and spherically symmetrical potentials. That is, the scattering amplitude is known at a fixed energy and we have to identify uniquely the potential. In Newton [20], Ch. 20.4 constructions are given to suggest that there exist potentials oscillating and decaying at infinity at the rate r −3/2 producing no scattering whatsoever, that is, for which all physical phase shifts δ n = δ(n + 1/2) vanish. Thus uniqueness can fail for slowly decaying potentials. Regge [26] proved uniqueness if all the (nonphysical) phase shifts δ(ν) are known and formulated some hints how to prove uniqueness from the scattering amplitude. Loeffel [18] made rigorous some considerations of Regge and proved uniqueness from the scattering amplitude for the potentials of compact support. This result has been considerably strengthened by Ramm [24] who proved that knowledge of a very sparse subsequence of the phase shifts δ n is enough to ensure uniqueness: the sum of reciprocals of the indices of known phase shifts must be infinite. In [16] it is shown that this condition is also necessary. Martin and Targonski [19] showed uniqueness for Yukawa-like potentials (special analytic potentials of exponential decay). If we remove the condition of spherical symmetry of the potential, uniqueness from the fixed-energy scattering amplitude is proved in Henkin and Novikov [15] if the potential is exponentially decaying and has sufficiently small norm. This smallness condition has been removed later in Novikov [21]. For potentials of compact support uniqueness has been obtained independently by Novikov [22] and Ramm [23]. Weder [30] proved that if two potentials q 1 and q 2 have a decay of order |x| −3−ε and q 2 − q 1 has compact support and the fixed-energy scattering amplitudes are the same then q 1 = q 2 . Ramm and Stefanov [25] proved uniqueness for potentials decaying faster than any exponential. Weder and Yafaev [31] proved that uniqueness holds for C ∞ -potentials which are (for large |x|) finite linear combinations of homogeneous functions of order −ρ j , ρ j > 3. Finally remark that concerning 2D inverse scattering problems Grinevich [14] constructed transparent potentials (with no scattering at a given energy) of decay O(|x| −2 ).
Below we prove uniqueness for 3D spherically symmetrical potentials where instead of exponential decay we only suppose that rq(r) is integrable at infinity: Then the scattering amplitude (7) uniquely determines q(r) a.e.

Proof of the results about the Krein function.
In what follows we borrow some ideas and notations from Loeffel [18]. Let ψ ± (r, ν) be the solutions of τ ψ = −ν 2 ψ such that By the quantities in (4) define the functions Let further We see that v = 0 if and only if ϕ = cψ for the function ψ in Theorem 1.2, that is, if λ = −ν 2 is an eigenvalue of the operator τ with domain (5). The Green function of τ , i.e. the kernel of (τ − λ) −1 , is see [18]. c.
As in [18], introduce the function Proof. From and then Now a subtraction gives Proof of Theorem 1.1 By (26) the operator (τ 2 + ν 2 ) −1 − (τ 1 + ν 2 ) −1 is of rank one, Thus there exists a spectral shift function ξ θ1,θ2 (t) such that we get that The function H(z) is Herglotz since m 1 is Herglotz and Consequently by the Aronszajn-Donoghue theorem [4] there is a measurable function 0 ≤ ξ(t) ≤ 1 with
Finally consider the uniqueness claim in Theorem 1.1. We need Lemma 2.4. The eigenvalues of the operators τ 1 and τ 2 separate each other. Moreover, if t < 0 increases, the function ξ θ1,θ2 (t) jumps from 0 to 1 at the eigenvalues of τ 1 , from 1 to 0 at the eigenvalues of τ 2 and stays constant in between.
Thus from d dz ln H(z) we get H(z) and then by (30) we obtain m 1 (z) and finally e 2iδ(ν) . By the uniqueness theorem of Regge [26] (see also [18]) the knowledge of e 2iδ(ν) , ν > 0 implies the knowledge of q a.e. under the conditions (2) and (8). This completes the proof of Theorem 1.1.
Proof of Theorem 1.3 Taking into account the previous proof, this is a simple transformation of Theorem 6.1 in Rybkin [27].

SPECTRAL SHIFT FUNCTIONS IN SCATTERING 11
From the estimate (42) below and from (8), (18) we get that In the first case consider the identity [5], 7.7.4(30) Taking into account the estimate the above considerations prove (13). Now consider the formula [29], 13.22 (2) which proves (14). The formula (15) can be similarly proved on the basis of Proof. We have to show that a 0 r 2ν+1 (q(r) − q(a − 0)) dr Take the number 0 < δ < a with (a − δ) 2ν ν 2 = a 2ν . Then δ → 0 as ν → +∞. Now  2)) we see that which implies (17) 4. Proof of Theorem 1.5. We will apply the Regge uniqueness theorem in [18], mentioned in the proof of Theorem 1.1 stating that the knowledge of e 2iδ(ν) , ν > 0 implies uniqueness of q under the conditions (2) and (8). The phase shifts δ(n+1/2) can be expressed from the scattering amplitude, hence we have to reconstruct the function δ(ν) from its values at ν = n+1/2. It is possible only if we have information about the growth of δ(ν) for large ν. We need the following The constants c = c(D) are independent of ν and r.
Taking into account (46) and the fact that δ(ν) = δ(ν) we see that δ(ν) is of polynomial growth on the half-plane ν ≥ 1/D. The Carlson uniqueness theorem (see e.g. in Fuchs [8]) says that if g(ν) is regular in ν ≥ 1/D and |g(ν)| ≤ c exp(a|ν|) for some a < π then the values g(n + 1/2), n ≥ n 0 uniquely determine g(ν). Since the knowledge of the scattering amplitude means the knowledge of δ(n + 1/2), we get δ(ν) by the Carlson theorem and then q by the Regge uniqueness theorem, mentioned earlier. This completes the proof.