An Approximate Method for Solving the Inverse Scattering Problem With Fixed-Energy Data

8/6/2019 An Approximate Method for Solving the Inverse Scattering Problem With Fixed-Energy Data

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J. . Ill-Posed Problems, Vol. 7, No. 6, pp. 561-571 (1999) VSP 1999

An approximate method for solving the inversescattering problem with fixed-energy dataA. G. Ramm* and W. Scheid f

Received M ay 12, 1999Abstract Assume that the potential < ? ( r ) , r.> 0, is known fo r r > > 0, and thephase shifts 6i(k) are known at a fixed energy, that is at a fixed fc, fo r / = 0,1,2,The inverse scattering problem is: find q(r) on the interval 0< r 0, that is, at a fixedenergy, is of interest in many areas of physics and engineering. A parameter-fitting procedure fo r solving this problem w as proposed in the early sixties byR. G. Newton an d discussed in [4, 2]. This procedure hats principal drawbackswhich have been discussed in detail in the paper [1].

If k > 0 is large, then one can use a stable numerical inversion of the fixed-energy scattering data in the Born approximation [10, Section 5.4]. Error esti-mates for the Born inversion are obtained in [10].

A n exact (mathematically rigorous) inversion method fo r solving a 3D in-verse scattering problem with fixed-energy noisy data is developed in [11] whereerror estimates were derived and the stability of the solution with respect tosmall perturbations of the data was estimated. This method, although rigor-ous, is not very simple.The aim of this paper is to propose a novel, approximate, quite simple in* Mathematics Department, Kansas State University, Manhattan, KS66506-2602, USA.

E-mail: [email protected]^Insti tut f r Theoretische Physik der Justus-Liebig-Universit t Giessen, Heinrich-Buff-Ring 16, D 35392, Giessen, Germany. E-mail: [email protected] work was supported by DAAD.


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562 A. G. Ramm and W. Scheid

principle, method for inverting the data { < 5 / } , / = 0,1,2,..., given at a fixedk > 0, for the potential q(r). In most physical problems one may assumethat q(r) is known for r > a (near infinity) , and one wants to find q(r) on theinterval [, ], where a > 0 is some known radius.Our basic idea is quite simple: since and q(r) for r > a are k n o w n , onecan easily compute the physical wave function () for r > a, and so the data

{(),'(}}, 1 = 0,1,2,. . . (1.1)can be obtained in a stable and numerically efficient way (described in Section 2below) from the original data {/}, {q(r) for r > a}.N ow we want to find q(r) on the interval r G [, ] from the data (1.1). Thefunction (] on the interval [0,oo ) solves the integral equationra(f\\ I -\) } (T) / Q i ( T ) p)q(p)(p) dp, ^ ^ (1.2)Jowhere g\ is defined in (A.12) and (r) is a known function which is writtenexplicitly in formula (2.7) below. Let

The numbers & / and are known. Taking r = in (1.2) and approximating by ' under the sign oi the integral one gets:

f a q ( p ) f i ( p ) d p = bli 1 = 0,1,2,3,... (1.4)./owhere

Differentiate (1.2) with respect to r, set r = a, and again replace by \under the sign of the integral. The result is:

wheredr

We have replaced (] in (1.2) under the sign of the integral by ^/ (p)- Such anapproximation is to some extent similar to the Born approximation and can bejustified if q(p) is small or / is large or is small. Note that this approximation isalso different from the Born approximation since * is defined by fo rmula (2.7)and incorporates the information about q(r) on the interval r > a.For potentials which are not small and for / not too large such an approxima-tion cannot be justified theoretically, but may still lead to acceptable numerical


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Inverse scattering problem 563

results since the error of this approximation is averaged in the process of inte-gration.We have derived approximate equations (1.4) and (1.6). Prom these equa-tions one can find an approximation to q(p) numerically. These equations yielda moment problem which has been studied in the literature. In [5] and [10, Sec-tion 6.2] a quasioptimal numerical method is given for solving moment problemswith noisy data.Note that the functions si (p ) differ only by a factor independent of f romthe funct ions fi(p) defined in (1.5). This is clear f rom the definition of thesefunct ions and formula (A . 12) for # /. Therefore one can use equations (1.4)fo r the recovery of q(r), and equations (1.6) are not used below. Prom thedefinition o f / / ( p ) it follows that the set {fi(p)}o


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564 A. G. Ramm and W. Scheid

For (1.11) to hold, we calculate i / /(r) and /() from the conditions:

(1.12)

(1.13)The parameter r G [0, ] in (1.12), (1.13) is arbitrary but fixed. Condition (1.12)is the normalization condition, condition (1.13) is the optimality condition forthe delta-sequence At,(r,p). It says that Ai(r,p) is concentrated near r = p,that is, AL is small outside a small neighborhood of the point = r. Onecan take 7 = 2 in (1.13) for example. The choice of 7 defines the degree ofconcentration of AL(T,P) near = r. We have taken \Ai\2 in (1.13) because inthis case the minimization problem (1,12),.(1.13).can.be reduced to solving alinear algebraic system of equations.

In order to calculate


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where q(r) is known. The solution to equation (2.1) is used in equation (2.7) todetermine } '.

The function / > 0 / is the uniqu e solution to the equation^oi + *Voi--^^=0, r > 0 (2.2)

which has the same asymptotics as as r - +00:' ~ e i< 5 ' sin ( f c r - + i/), r -> oo. (2.3)

This formula is derived in Appendix (formula (A .29)). Let us derive a formulafor ^o/ If has the asymptotics (2.3), then

= ciii / (Ar) + C2Vi(kr) ~ c\ sin (A T ) - 0 2 cos ( f c r ) , r -> oo (2.4)where it/ and vi ar e defined in (A.I) and c i , C 2 ar e some constants.Prom (2.4) and the asymptotics (2.3) for / > 0 / , it follows that

d = e i < 5 i cos(A/), c2--e i < 5 i sin(i,). (2.5)Thus we obtain an explicit formula for t / > o / :

^01 (Ar) = e i < 5 / cos(i/) u/(fcr) - e i < 5 / sin(i/) vi(kr). (2.6)

2.2. Calculating (0)Let us find ^ in equation (1.2). We start with the integral equation (A. 14)for the physical wave function and rewri te (A .14) as equation (1.2) with

V,(0) = ri\r,k) :=Ul(kr) - 9l(r,p)q(p)^(p,k)dp. (2.7)Ja

Formula (2.7) solves the problem of finding in equation (1.2). Indeed, inSection 2.1 w e have sh ow n how to calculate (, k ) fo r > , so that the right-hand side of (2.7) is now a know n function of r. This completes the descriptionof the numerical aspects of the proposed inversion me thod.3. SUMM ARY OF THE PROPOSED METHODLet us summarize the method:

1. Given the data {/, / = 0 , 1 , 2 , . . . ; (?(r), r > a} one calculates ? / > o / byformula (2.6), solves equation (2.1) by iterations for r > , and obtainsthe data (1.1).


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566 A. G. Ramm and W. Scheid2. Given the data (1.1), one calculates b i by formulas (1.3), then choosesan integer L and solves the moment problem (1.4) for 0 < / < L. Theapproximate solution


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Inverse scattering problem 567APPENDIX AEquation (2.2) has linearly independent solutions ui(kr) and t > / ( f c r ) , called Ric-cati-Bessel functions:/ 7 / 7 7 *ui(r) := W yJi+i/2(r) := rj/(r), v / ( r ) = W yAT/+1/2(r) := rn/(r) (A.I)where ( ) and N,/(r) are the Bessel and N eum ann functions regular and irreg-ular at the origin respectively, ji and H I are the spherical Bessel and Neumannfunctions. One has

vi(r) ~ - cos(r ) , u / ( r ) ~ sin(r ) , r -) oo (A.2)(2/-1)!! rm**(r)~-V / ;", ^ ( - 1 ^ (A'3)

The regular solution to (2.2), denoted by < p o i ( k r ) and defined by the condi-tion-* (.4)(2)! ! 'is given by the formula

(A.5)which follows from (A.3) an d (A.4).The Jost solution /o / to (2.2) defined by the conditione i f c r , r ->oo (A.6)

is given by the formulafQl(kr) = el l /2ui(kr) - eilir/Vfcr)-ie i / 7 r/2( i z / 4- it;,) (A.7)

as follows from (A.2).The Wronskian isF < * i ( k ) := W [ f a , < P o i ] = f o i ( k r ) - < p Q l ( k r ) - ^(*)/(*) = (.8)dr dr 1

as follows from (A.3), (A.4) , (A.7), and the fact that this Wronskian does notdepend on r, so it can be calculated at r = 0.In equation (2.1)we use the Green function f / ( r , p ) satisfying the equation4+*6 ( , ) - 6(r,p)=-i(r- ,) (.9)

where i(r p) is the delta-function, and vanishing for < r:kr)], P>r, < r.


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Inverse scattering problem 569Let r - -foe in (A.14). Then (A.2), (A.5), (A.6), (A.8), and (A.12) imply

(1+1)/2

~ - [e~lkr - el*lelhrSi], r -> -f oo (A.19)Z twhere 9 /%oS/ = 1 -f / ui(kp)q(p)il>i(p,k)dp. (A.20) JoThe scattering amplitude (',, J k ) defined by the formula

= exp(ifca x) -f A ( a ' , a , f c ) l o(-), r := |x| - oo, r = a' (A.21)can be written as

If q = q(p), := |y|, then00

where ^A / ( f e ) = - T J / ui(kp)q(p)ij}i(pjk)Ap (A.24)

as one gets after su bs tituting the complex conjugate of (A.16), (A.17), and (A.23)into (A.22) . From (A.20) and (A.24) one getsSi = 1 - - (.25)

which is the fundamental relation between the 5-matrix and the scatteringamplitude:5=7-JU (A.26)

Here 5, /, and A are operators in L2(52), / is the identity operator. Since theoperator 5 is unitary in L2(52) and Si are the eigenvalues of 5 in the eigenbasisof spherical harmonics in the case of spherically symmetric potentials, one has15/| = 1, so, for some real number one writes5f = e2ii|. (A.27)

The numbers normalized by the conditionJ / ( f c ) - > 0 a s & - + o o (A.28)

are called the phase shifts. They are in one-to-one correspondence with thenumbers Si if (A.28) is assumed. We do assume (A.2 8).


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570 A. G. Ramm and W. ScheidFormula (A.19) can be rewritten as

~ e101 : = e 1 < 5 / sin (kr h i / ) , r -> oo.(A.29)Formula (A.29) was used in Section 2 (see formu la (2.3)) .Note that (A.25) and (A.27) imply = y e i < 5 /s i n ( 5 / ) . (A.30)

This formula and (A.23) show that if q(x) = ^( |x | )5 then th e information whichis given by the knowledge of the fixed-energy phase shifts {/, / = 0 , 1 , 2 , . . . }is equivalent to the inform ation w hich is given by the know ledge of the fixed-energy scattering amplitude (',) where a7 and run through th e whole ofth e unit sphere S2. By Ramm's uniqueness theorem [6], it follows that if thefixed-energy phase shifts corresponding to a compactly supported potential q(r)are known for / = 0 , 1 , 2 , 3 , . . . , then th e potential q(r) is uniquely defined, andtherefore th e phase shifts are uniquely defined for all / on the complex planeby their values for / running through th e integers only: / = 0 , 1 , 2 , AcknowledgementThis paper w as written while A , G , Ramm w as visiting the Institute of The-oretical Physics at the University of Giessen. A GR thanks D A A D and thisUniversity for hospitality.REFERENCES

1. R. Airapetyan, A . G . Ramm, and A. Smirnova, Ex ample of two differentpotentials wh ich have practically th e same fixed-energy phase shifts. Phys.Lett. (1999) A254, No. 3-4, 141-148.2 . K. Chadan and P. Sabatier, Inverse P roblems of Quantum Scattering The-ory. Springer Verlag, New York, 1989.

3. M. M nchow and W. Scheid, M odification of the Newton method for theinverse scattering problem at fixed energy. Phys. Rev. Lett. (1980) 44,1299-1302.4. R. Newton, Scattering of Waves and Particles. Springer Verlag, New

York, 1982.5. A. G. Ramm, Optimal estimation from limited noisy data. J. Math. AnalAppl. (1987) 125, 258-266.6. A. G. Ramm, Recovery of the potential from fixed energy scattering data.Inverse Problems (1988) 4, 877-886; (1989) 5, 255 (Corr igendum).


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7. A. G. Ram m, M ultidimensional inverse problems and completeness of theproducts of solutions to PDE. J. Math. Anal. Appl. (1988) 134, 211-253;(1989) 139, 302 (Addenda).8. A. G. Ram m , Uniqueness theorems for multidimensional inverse problemswith unbounded coefficients. J. Math. Anal Appl. (1988) 136, 568-574.9. A. G. Ramm, Completeness of the products of solutions of PDE and in-verse problems. Inverse Problems (1990) 6, 643-664.

10. A. G. Ramm, Multidimensional Inverse Scattering Problems. Longman/Wiley, New York, 1992.11. A . G . Ramm, Stability estimates in inverse scattering. Acta Appl. Math.(1992)28, No. 1, 1-42.12. A. G. Ramm and A .B . Smirnova, A numerical method for solving theinverse scattering problem w ith fixed-energy phase shifts. J. Inv. Ill-PosedProblems (to appear).


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An Approximate Method for Solving the Inverse Scattering Problem With Fixed-Energy Data