J. Inv. I-Posed Probems, Vo. 7, No. 6, pp. 561 571 (1999) c VSP 1999 An approximate method for soving the inverse scattering probem with fixed-energy data A. G. Ramm and W. Scheid Received May 12, 1999 Abstract Assume that the potentia q(r), r >, is known for r a >, and the phase shifts δ (k) are known at a fixed energy, that is at a fixed k, for =, 1, 2,.... The inverse scattering probem is: find q(r) on the interva r a, given the above data. A very simpe approximate numerica method is proposed for soving this inverse probem. The method consists in reduction of this probem to a moment probem for q(r) on the interva r [, a]. This moment probem can be soved numericay. 1. INTRODUCTION Finding a potentia q(r), r = x, from the phase shifts δ (k) for anguar quantum numbers =, 1, 2,..., known at a fixed wave number k >, that is, at a fixed energy, is of interest in many areas of physics and engineering. A parameterfitting procedure for soving this probem was proposed in the eary sixties by R. G. Newton and discussed in [4, 2]. This procedure has principa drawbacks which have been discussed in detai in the paper [1]. In [1] AGR pointed out for the first time that the Newton Sabatier method widey used by the physicists to invert fixed-energy phase shifts for the potentia is not reay an inversion method but a parameter-fitting procedure; that this procedure is not aways appicabe to the data; and that it is not proved that this procedure eads to the origina potentia, which produced the origina phase shifts. Moreover, the Newton Sabatier procedure, if it is appicabe, can ony produce a potentia which is anaytic in a neighborhood of (, + ), with a possibe poe at r =. A genera numerica method is given in [1] to construct piecewise-constant compacty supported sphericay-symmetric potentias which Mathematics Department, Kansas State University, Manhattan, KS6656-262, USA. E-mai: ramm@math.ksu.edu Institut für Theoretische Physik der Justus-Liebig-Universität Giessen, Heinrich-Buff- Ring 16, D 35392, Giessen, Germany. E-mai: werner.scheid@theo.physik.uni-giessen.de The work was supported by DAAD.
562 A. G. Ramm and W. Scheid are quite different but have neary the same sets of phase shifts at a fixed energy. If k > is arge, then one can use a stabe numerica inversion of the fixedenergy scattering data in the Born approximation [1, Section 5.4]. Error estimates for the Born inversion are obtained in [1]. An exact (mathematicay rigorous) inversion method for soving a 3D inverse scattering probem with fixed-energy noisy data is deveoped in [11] where error estimates were derived and the stabiity of the soution with respect to sma perturbations of the data was estimated. This method, athough rigorous, is not very simpe. The aim of this paper is to propose a nove, approximate, quite simpe in principe, method for inverting the data {δ }, =, 1, 2,..., given at a fixed k >, for the potentia q(r). In most physica probems one may assume that q(r) is known for r a (near infinity), and one wants to find q(r) on the interva [, a], where a > is some known radius. Our basic idea is quite simpe: since δ and q(r) for r a are known, one can easiy compute the physica wave function ψ (r) for r a, and so the data {ψ (a), ψ (a)}, =, 1, 2,... (1.1) can be obtained in a stabe and numericay efficient way (described in Section 2 beow) from the origina data {δ }, {q(r) for r a}. Now we want to find q(r) on the interva r [, a] from the data (1.1). The function ψ (r) on the interva [, ) soves the integra equation ψ (r) = ψ () (r) g (r, ρ)q(ρ)ψ (ρ) dρ, r a (1.2) where g is defined in (A.12) and ψ () (r) is a known function which is written expicity in formua (2.7) beow. Let [ψ (a) ψ () (a)] := b, [ψ (a) ψ () (a)] := β, =, 1,.... (1.3) The numbers b and β are known. Taking r = a in (1.2) and approximating ψ by ψ () under the sign of the integra one gets: where q(ρ)f (ρ) dρ = b, =, 1, 2, 3,... (1.4) f (ρ) := g (a, ρ)ψ () (ρ). (1.5) Differentiate (1.2) with respect to r, set r = a, and again repace ψ by ψ () under the sign of the integra. The resut is: where q(ρ)s (ρ) dρ = β, =, 1, 2, 3,... (1.6) s (ρ) := g (r, ρ) r ψ () (ρ). (1.7) r=a
Inverse scattering probem 563 We have repaced ψ (ρ) in (1.2) under the sign of the integra by ψ () (ρ). Such an approximation is to some extent simiar to the Born approximation and can be justified if q(ρ) is sma or is arge or a is sma. Note that this approximation is aso different from the Born approximation since ψ () is defined by formua (2.7) and incorporates the information about q(r) on the interva r a. For potentias which are not sma and for not too arge such an approximation cannot be justified theoreticay, but may sti ead to acceptabe numerica resuts since the error of this approximation is averaged in the process of integration. We have derived approximate equations (1.4) and (1.6). From these equations one can find an approximation to q(ρ) numericay. These equations yied a moment probem which has been studied in the iterature. In [5] and [1, Section 6.2] a quasioptima numerica method is given for soving moment probems with noisy data. Note that the functions s (ρ) differ ony by a factor independent of ρ from the functions f (ρ) defined in (1.5). This is cear from the definition of these functions and formua (A.12) for g. Therefore one can use equations (1.4) for the recovery of q(r), and equations (1.6) are not used beow. From the definition of f (ρ) it foows that the set {f (ρ)} L of these functions is ineary independent for any finite positive integer L. Aso, one can prove that the moment probem (1.4) has at most one soution. Indeed, the corresponding homogeneous probem (1.4), corresponding to b = for a =, 1, 2,..., has ony the trivia soution because the set of functions {f (ρ)} is compete in L 2 (, a) as foows from the resut in [6] (see aso [1]). To see this, note that the function {f (ρ)} < differs ony by a factor independent of ρ from the function u 2 (ρ) (the functions u are defined in (A.1)), and the set of these functions is the set of products of soutions to homogeneous equation (2.2) (in Section 2) for a =, 1, 2,.... The set of these products is compete in L 2 (, a) because the set of functions {u 2 (ρ)ρ 2 Y (α)y (β)} =,1,2,..., α, β S 2, where S 2 is the unit sphere in R 3, is the set of products of soutions to the homogeneous Schrödinger equation which is compete in L 2 (B a ), as foows from the resuts in [1]. Let us expain the idea of the method discussed in detai in [5], which is simiar to the we-known Backus Gibert method [1]. Fix a natura number L and ook for an approximation of q(r) of the form q L (r) := L b ν (r) := = A L (r, ρ)q(ρ) dρ (1.8) where the kerne A L is defined by formua (1.1) beow, b are the known numbers given in (1.3), and ν (r) are not known and shoud be found for any fixed r [, a] so that q L (r) q(r) as L. (1.9) The norm in (1.9) is L 2 [, a] or C[, a] norm depending on whether q L 2 [, a]
564 A. G. Ramm and W. Scheid or q C[, a]. Condition (1.9) hods if the sequence of the kernes is a deta-sequence, that is, A L (r, ρ) := L ν (r)f (ρ) (1.1) = A L (r, ρ) δ(r ρ), L + (1.11) where δ(r ρ) is the deta-function. For (1.11) to hod, we cacuate ν (r) and µ (r) from the conditions: A L (r, ρ) dρ = 1 (1.12) A L (r, ρ) 2 r ρ γ dρ = min. (1.13) The parameter r [, a] in (1.12), (1.13) is arbitrary but fixed. Condition (1.12) is the normaization condition, condition (1.13) is the optimaity condition for the deta-sequence A L (r, ρ). It says that A L (r, ρ) is concentrated near r = ρ, that is, A L is sma outside a sma neighborhood of the point ρ = r. One can take γ = 2 in (1.13) for exampe. The choice of γ defines the degree of concentration of A L (r, ρ) near ρ = r. We have taken A L 2 in (1.13) because in this case the minimization probem (1.12), (1.13) can be reduced to soving a inear agebraic system of equations. In order to cacuate q L (r) by formua (1.8) one has to cacuate ν (r) and µ (r) for different vaues of r [, a]. The probem (1.12), (1.13) is a probem of minimization of the quadratic form (1.13) with respect to the variabes ν (r) (which are considered as numbers for a fixed r) under the inear constraint (1.12). Such a probem can be soved, for exampe, by the Lagrange mutipiers method [1, Section 6.2]. In Section 2 we discuss numerica aspects of the proposed method for soving the inverse scattering probem. The idea of the method is aso appicabe to the inverse scattering probem with data given at a fixed for a k >, or for some vaues of k. We specify ψ () in equation (1.2) and give a method for computing the data (1.1). In Section 3 a summary of the proposed method is given. In the Appendix we have coected a the necessary reference formuas in order to make this paper sef-contained. In [12] numerica resuts obtained by the proposed approximate inversion method are given. 2. NUMERICAL ASPECTS
Inverse scattering probem 565 2.1. Cacuating ψ and the data (1.1) We start with a method for cacuating the data (1.1) from the origina data {δ, =, 1, 2,...} and {q(r), r a}. The foowing integra equation is convenient for finding the data (1.1): ψ (r) = ψ (r) r ξ (r, ρ)q(ρ)ψ (ρ) dρ, r > a (2.1) where the Green function ξ (r, ρ) is defined in (A.1) and ψ is defined by formua (2.6) beow. Equation (2.1) is a Voterra integra equation. It can be soved staby and numericay efficienty by iterations on the interva r a where q(r) is known. The soution to equation (2.1) is used in equation (2.7) to determine ψ (). The function ψ is the unique soution to the equation ψ + k 2 ( + 1) ψ r 2 ψ =, r > (2.2) which has the same asymptotics as ψ as r + : ψ e iδ sin ( kr π 2 + δ ), r. (2.3) This formua is derived in Appendix (formua (A.29)). Let us derive a formua for ψ. If ψ has the asymptotics (2.3), then ψ = c 1 u (kr) + c 2 v (kr) c 1 sin ( kr π 2 ) c2 cos ( kr π 2 where u and v are defined in (A.1) and c 1, c 2 are some constants. From (2.4) and the asymptotics (2.3) for ψ, it foows that ), r (2.4) c 1 = e iδ cos(δ ), c 2 = e iδ sin(δ ). (2.5) Thus we obtain an expicit formua for ψ : ψ (kr) = e iδ cos(δ ) u (kr) e iδ sin(δ ) v (kr). (2.6) 2.2. Cacuating ψ () Let us find ψ () in equation (1.2). We start with the integra equation (A.14) for the physica wave function ψ and rewrite (A.14) as equation (1.2) with ψ () = ψ () (r, k) := u (kr) a g (r, ρ)q(ρ)ψ (ρ, k) dρ. (2.7) Formua (2.7) soves the probem of finding ψ () in equation (1.2). Indeed, in Section 2.1 we have shown how to cacuate ψ (ρ, k) for ρ a, so that the righthand side of (2.7) is now a known function of r. This competes the description of the numerica aspects of the proposed inversion method.
566 A. G. Ramm and W. Scheid 3. SUMMARY OF THE PROPOSED METHOD Let us summarize the method: 1. Given the data {δ, =, 1, 2,... ; q(r), r a} one cacuates ψ by formua (2.6), soves equation (2.1) by iterations for r a, and obtains the data (1.1). 2. Given the data (1.1), one cacuates b by formuas (1.3), then chooses an integer L and soves the moment probem (1.4) for L. The approximate soution q L (r) is given by formua (1.8) in which ν (r) are found by soving the optimization probem (1.12), (1.13). In concusion we make severa remarks. Remark 1. This method can be formuated aso for the case when the data b of the moment probem are known with some random errors (see [1, Section 6.2] for detais). This idea can aso be tried for a numerica inversion of fixed-energy scattering data. Remark 2. This inverse scattering probem with fixed-energy data is highy i-posed if k is not very arge. Specific estimates of the conditiona stabiity of the soution to this probem were obtained in [11] and iustrated in [1] (see aso [1]). Athough it was proved in [6] (see aso [7 9]) that the 3D inverse scattering probem with fixed-energy scattering data has at most one soution in the cass of compacty supported potentias, but the estimates of the stabiity of this soution with respect to perturbation of the scattering data obtained in [11] indicate that the stabiity is very weak: theoreticay one has to have the scattering data known with very high accuracy in order to recover the potentia with a moderate accuracy. Of course the estimates in [11] cover the worst possibe case, and in practice the recovery may be much better than the error estimates from [11] guarantee. In a recent paper [1] an exampe is given of two quite different piecewiseconstant, rea-vaued, compacty supported potentias which have practicay the same phase shifts at a fixed energy for a (the phase shifts differ by a quantity of order 1 5 ). This resut shows that it is necessary to have some a priori information about the unknown potentia q(r) in order to be abe to recover it with some accuracy. Remark 3. In the Born approximation, inversion of fixed-energy scattering data is reduced to soving an integra equation of the form: B a q(x)e iξ x dx = f(ξ), ξ 2k (3.1) where k > is a fixed given number, x R 3, and a > is the radius of a ba B a outside of which q(x) =. If one knows f(ξ) for a ξ R 3, one can find q(x) by taking the inverse Fourier transform of f(ξ). If f(ξ) is known ony for ξ < 2k,
Inverse scattering probem 567 one can sti recover a compacty supported q(x) uniquey and anayticay with an arbitrary accuracy, if f(ξ) is known exacty for ξ 2k (see [1, Section 6.1], where anaytic inversion formuas for (3.1) are derived). Remark 4. A numerica impementation of the parameter-fitting procedure of R. G. Newton requires a priori information about the potentia q, in particuar, the knowedge of q for r a, see [3]. APPENDIX A Equation (2.2) has ineary independent soutions u (kr) and v (kr), caed Riccati Besse functions: πr πr u (r) := 2 J +1/2(r) := rj (r), v (r) = 2 N +1/2(r) := rn (r) (A.1) where J ν (r) and N ν (r) are the Besse and Neumann functions reguar and irreguar at the origin respectivey, j and n are the spherica Besse and Neumann functions. One has v (r) cos(r π 2 ), u (r) sin(r π ), r (A.2) 2 (2 1)!! v (r) r, u (r) r+1, r. (A.3) (2 + 1)!! The reguar soution to (2.2), denoted by ϕ (kr) and defined by the condition ϕ (kr) r+1 (2 + 1)!!, r (A.4) is given by the formua ϕ (kr) = u (kr) k +1 which foows from (A.3) and (A.4). The Jost soution f to (2.2) defined by the condition is given by the formua (A.5) f (kr) e ikr, r (A.6) f (kr) = e i(+1)π/2 u (kr) e iπ/2 v (kr) = ie iπ/2 (u + iv ) (A.7) as foows from (A.2). The Wronskian is F (k) := W [f, ϕ ] = f (kr) d dr ϕ (kr) ϕ (kr) d dr f (kr) = eiπ/2 k (A.8) as foows from (A.3), (A.4), (A.7), and the fact that this Wronskian does not depend on r, so it can be cacuated at r =.
568 A. G. Ramm and W. Scheid In equation (2.1) we use the Green function ξ (r, ρ) satisfying the equation d 2 ξ (r, ρ) dr 2 + k 2 ( + 1) ξ (r, ρ) r 2 ξ (r, ρ) = δ(r ρ) (A.9) where δ(r ρ) is the deta-function, and vanishing for ρ < r: { e iπ/2 k [ ϕ (kρ)f (kr) + f (kρ)ϕ (kr)], ρ r ξ (r, ρ) =, ρ < r. (A.1) Indeed, ξ soves (A.9) for r ρ since, as a function of r, it is a inear combination of soutions to the homogeneous equation (A.9). Moreover, dξ (r, ρ) dr r=ρ+ r=ρ by (A.8). Thus (A.9) foows. The Green function = e iπ/2 k [ ϕ (kρ) df (kr) dr g (r, ρ) := { F 1 (k)ϕ (kρ)f (kr), r ρ F 1 (k)ϕ (kr)f (kρ), r < ρ f (kρ) dϕ ] (kr) = 1 dr r=ρ (A.11) (A.12) (where F (k) is defined in (A.8), F = e iπ/2 k ) soves (A.9) and satisfies the radiation condition at infinity: g r ikg as r +. (A.13) Indeed, (A.13) foows from (A.12) and (A.6), and to verify that g soves equation (A.9) one argues in the same way as was done after formua (A.1). The integra equation for the physica wave function ψ (r, k) is: ψ (r, k) = u (kr) g (r, ρ)q(ρ)ψ (ρ, k) dρ. This equation foows from the standard three-dimensiona equation: (A.14) ψ(x, α, k) = ψ (x, α, k) R3 e ik x y q(ρ)ψ(y, α, k) dy, 4π x y where ψ (x, α, k) := e ikα x, and α S 2 is a given unit vector. If one writes e ikα x = 4π u (kr) k i Y (x )Y (α), r x := x x ρ = y, r = x (A.15) (A.16) ψ = 4π ψ (r, k) k i Y (x )Y (α) r (A.17)
Inverse scattering probem 569 e ik x y 4π x y = g (r, ρ) Y (x )Y (y rρ ) (A.18) where Y := Y m, m, are the orthonorma spherica harmonics, and inserts (A.16) (A.18) into (A.15), then one gets (A.14). In (A.16) (A.18) the summation with respect to, =, 1, 2, 3,..., incudes aso the summation with respect to m, m. Let r + in (A.14). Then (A.2), (A.5), (A.6), (A.8), and (A.12) impy where ψ eiπ(+1)/2 [e ikr e iπ e ikr S ], r + (A.19) 2 S = 1 + 2 ik u (kρ)q(ρ)ψ (ρ, k) dρ. The scattering ampitude A(α, α, k) defined by the formua (A.2) ψ = exp(ikα x) + A(α, α, k) eikr r can be written as A(α, α, k) = 1 4π If q = q(ρ), ρ := y, then + o ( 1) x, r := x, r x = α R 3 e ikα y q(ρ)ψ(y, α, k) dy. (A.21) (A.22) where A(α, α, k) = A (k) = 4π k 2 A (k)y (α )Y (α) = u (kρ)q(ρ)ψ (ρ, k) dρ (A.23) (A.24) as one gets after substituting the compex conjugate of (A.16), (A.17), and (A.23) into (A.22). From (A.2) and (A.24) one gets S = 1 k 2πi A (A.25) which is the fundamenta reation between the S-matrix and the scattering ampitude: S = I k 2πi A. (A.26) Here S, I, and A are operators in L 2 (S 2 ), I is the identity operator. Since the operator S is unitary in L 2 (S 2 ) and S are the eigenvaues of S in the eigenbasis of spherica harmonics in the case of sphericay symmetric potentias, one has S = 1, so, for some rea number δ one writes S = e 2iδ. (A.27)
57 A. G. Ramm and W. Scheid The numbers δ normaized by the condition δ (k) as k (A.28) are caed the phase shifts. They are in one-to-one correspondence with the numbers S if (A.28) is assumed. We do assume (A.28). Formua (A.19) can be rewritten as ψ e iδ ei(kr π/2+δ ) e i(kr π/2+δ ) 2i Formua (A.29) was used in Section 2 (see formua (2.3)). Note that (A.25) and (A.27) impy = e iδ sin ( kr π 2 + δ ), r. (A.29) A = 4π k eiδ sin(δ ). (A.3) This formua and (A.23) show that if q(x) = q( x ), then the information which is given by the knowedge of the fixed-energy phase shifts {δ, =, 1, 2,... } is equivaent to the information which is given by the knowedge of the fixedenergy scattering ampitude A(α, α) where α and α run through the whoe of the unit sphere S 2. By Ramm s uniqueness theorem [6], it foows that if the fixed-energy phase shifts corresponding to a compacty supported potentia q(r) are known for =, 1, 2, 3,..., then the potentia q(r) is uniquey defined, and therefore the phase shifts δ are uniquey defined for a on the compex pane by their vaues for running through the integers ony: =, 1, 2,.... Acknowedgement This paper was written whie A. G. Ramm was visiting the Institute of Theoretica Physics at the University of Giessen. AGR thanks DAAD and this University for hospitaity. REFERENCES 1. R. Airapetyan, A. G. Ramm, and A. Smirnova, Exampe of two different potentias which have practicay the same fixed-energy phase shifts. Phys. Lett. (1999) A254, No. 3 4, 141 148. 2. K. Chadan and P. Sabatier, Inverse Probems of Quantum Scattering Theory. Springer Verag, New York, 1989. 3. M. Münchow and W. Scheid, Modification of the Newton method for the inverse scattering probem at fixed energy. Phys. Rev. Lett. (198) 44, 1299 132. 4. R. Newton, Scattering of Waves and Partices. Springer Verag, New York, 1982.
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