# Quantum mechanical inverse scattering problem at fixed energy: a constructive method

**ABSTRACT** The inverse scattering problem of the three-dimensional Schroedinger equation

is considered at fixed scattering energy with spherically symmetric potentials.

The phase shifts determine the potential therefore a constructive scheme for

recovering the scattering potential from a finite set of phase shifts at a

fixed energy is of interest. Such a scheme is suggested by Cox and Thompson [3]

and their method is revisited here. Also some new results are added arising

from investigation of asymptotics of potentials and concerning statistics of

colliding particles. A condition is given [2] for the construction of

potentials belonging to the class L_1,1 which are the physically meaningful

ones. An uniqueness theorem is obtained [2] in the special case of one given

phase shift by applying the previous condition. It is shown that if only one

phase shift is specified for the inversion procedure the unique potential

obtained by the Cox-Thompson scheme yields the one specified phase shift while

the others are small in a certain sense. The case of two given phase shifts is

also discussed by numerical treatment and synthetic examples are given to

illustrate the results. Besides the new results this contribution provides a

systematic treatment of the CT method.

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**ABSTRACT:**In order to learn more about the precision of the inversion by the Cox–Thompson method, we investigated the inversion of phase shifts of a singular potential, namely of a Coulomb potential. Using asymptotically Riccati–Bessel functions as reference functions, we could only approximately reproduce the singularity of the Coulomb potential at the origin. We also show uncertainties in the inverted potential due to different minima in the minimization solution of the nonlinear equations of the Cox–Thompson procedure. As a result, we conclude that one has to take much care with the inversion of experimental phase shifts suffering from measurement errors.Journal of Physics A Mathematical and Theoretical 05/2010; 43(22):225302. · 1.77 Impact Factor - SourceAvailable from: psu.edu[Show abstract] [Hide abstract]

**ABSTRACT:**The Cox–Thompson inverse scattering method at fixed energy has been generalized to treat complex phase shifts derived from experiments. New formulae for relating phase shifts to shifted angular momenta are derived. The method is applied to phase shifts of known potentials in order to test its quality and stability and, further, it is used to invert experimental n-α and n-12C phase shifts.Journal of Physics G Nuclear and Particle Physics 05/2006; 32(6):849. · 2.84 Impact Factor - SourceAvailable from: Miklos Horvath[Show abstract] [Hide abstract]

**ABSTRACT:**Solution of the Cox-Thompson inverse scattering problem at fixed energy [1,2,3] is reformulated resulting in semi-analytic equations. The new set of equations for the normalization constants and the nonphysical (shifted) angular momenta are free of matrix inversion operations. This simplification is a result of treating only the input phase shifts of partial waves of a given parity. Therefore, the proposed method can be applied for identical particle scattering of the bosonic type (or for certain cases of identical fermionic scattering). The new formulae are expected to be numerically more efficient than the previous ones. Based on the semi-analytic equations an approximate method is proposed for the generic inverse scattering problem, when partial waves of arbitrary parity are considered.Modern Physics Letters B 11/2011; 22(23). · 0.69 Impact Factor

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arXiv:1111.6009v1 [math-ph] 25 Nov 2011

Quantum mechanical inverse scattering problem

at fixed energy: a constructive method

Tam´ as P´ almai∗and Barnab´ as Apagyi†

Department of Theoretical Physics

Budapest University of Technology and Economics

Budafoki ut 8., H-1111 Budapest, Hungary

November 28, 2011

Abstract

The inverse scattering problem of the three-dimensional Schr¨ odinger

equation is considered at fixed scattering energy with spherically symmet-

ric potentials. The phase shifts determine the potential therefore a con-

structive scheme for recovering the scattering potential from a finite set of

phase shifts at a fixed energy is of interest. Such a scheme is suggested by

Cox and Thompson [3] and their method is revisited here. Also some new

results are added arising from investigation of asymptotics of potentials

and concerning statistics of colliding particles. A condition is given [2] for

the construction of potentials belonging to the class L1,1 which are the

physically meaningful ones. An uniqueness theorem is obtained [2] in the

special case of one given phase shift by applying the previous condition.

It is shown that if only one phase shift is specified for the inversion proce-

dure the unique potential obtained by the Cox-Thompson scheme yields

the one specified phase shift while the others are small in a certain sense.

The case of two given phase shifts is also discussed by numerical treat-

ment and synthetic examples are given to illustrate the results. Besides

the new results this contribution provides a systematic treatment of the

CT method.

∗Electronic mail: palmai@phy.bme.hu

†Electronic mail: apagyi@phy.bme.hu

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1General results, introduction

We start with the Schr¨ odinger equation in R3at a fixed positive energy

[∇2+ 1 − q(x)]Ψ(x,α) = 0in R3

(1)

whose scattering solution takes the form

Ψ(x,α) = eix·α+ A(α′,α)eir

r

+ o

?1

r

?

,r = |x| → ∞,α′=x

r,

(2)

where α is the direction of the incident wave and A(α,α′) is the scattering

amplitude. The following general theorem is due to Ramm [8].

Theorem 1.1. A(α′,α) scattering amplitude ∀α′∈˜S2

small open subsets of S2) determine q(x) uniquely in the function class Qa=

{q : q = ¯ q, q(x), |x| > a, q(x) ∈ L2(Ba)}, Ba= {x : x ∈ R3, |x| < a}.

1, ∀α ∈˜S2

2(arbitrary

The present treatment is restricted to spherically symmetric potentials, that

is q(x) = q(r). In this case we have the partial wave expansion of the wave

function:

∞

?

where ψℓ(x) satisfies the radial Schr¨ odinger equation (see below) with the ap-

propriate boundary conditions.

For spherically symmetric potentials the scattering amplitude takes a similar

expansion form, namely

Ψ(x,α) =

ℓ=0

ℓ

?

m=−ℓ

4πiℓψℓ(r)

r

Yℓm(x/r)¯Yℓm(α),(3)

A(α′,α) = A(α′· α) =

∞

?

ℓ=0

ℓ

?

m=−ℓ

AℓYℓm(α′)¯Yℓm(α),

Aℓ= 2πi(1 − e2iδℓ) = 4πeiδℓsinδℓ.

The phase shifts, {δℓ}ℓ=0,1,2,...can be restricted to −π

the experimental point of view they are undetermined to an additive factor

of kπ, k ∈ Z. Then it is apparent that the knowledge of the phase shifts is

equivalent to that of the scattering amplitude. However not all the phase shifts

(as considered in the theorem of Loeffel [1]) are needed for unique reconstruction

of the potential. The following result is due to Horv´ ath and Ramm [9, 6].

2≤ δℓ ≤

π

2, since from

Theorem 1.2. The phase shifts {δℓ}ℓ∈Ldetermine the potential uniquely in the

class Q = {q : q = ¯ q, q(x) = q(r), r = |x|, q(r) = 0, r > a; rq(r) ∈ L1(0,a)} if

the M¨ untz condition holds for L:

?

ℓ∈L,ℓ?=0

1

ℓ= ∞.(4)

This condition is almost necessary in the sense that in the class Qσwith 0 < σ <

2 the set {δℓ} with?ℓ−1< ∞ is not enough to recover the potential uniquely.

Qσ= {q : q = ¯ q, q(x) = q(r), r = |x|, q(r) = 0, r > a; r1−σq(r) ∈ L1(0,a)}.

2

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It is because of this result that one may look for a potential when a part of

the scattering phase shifts are known.

The phase shift δℓappears in the asymptotic form of the regular solution of

the Schr¨ odinger equation, which is defined in the following manner:

Lrϕℓ(r) = ℓ(ℓ + 1)ϕℓ(r),Lr=

?

r2d2

dr2+ r2− r2q(r)

?

,

ϕℓ(r) =

rℓ+1

(2ℓ + 1)!!+ o(rℓ+1),r → 0,

ϕℓ(r) = Bℓsin(r −ℓπ

2

+ δℓ) + o(1),r → ∞

with Bℓ being a constant. Note that for the q ≡ 0 zero potential we have

Lr0uℓ(r) = ℓ(ℓ + 1)uℓ(r), where Lr0=

and Jn(r) is the nth order Bessel function of the first kind.

The Povzner-Levitan representation in the fixed energy problem [5] has long

been used, however the existence and uniqueness of a K(r,r′) ∈ C2(R+× R+)

transformation kernel have been proven rigorously only recently [6] for potential

q(r) ∈ C1(0,a) independent of ℓ and satisfying

?

r2 ∂2

∂r2+ r2?

and uℓ(r) =?πr

2Jℓ+1

2(r)

ϕℓ(r) = uℓ(r) −

?r

0

K(r,ρ)uℓ(ρ)ρ−2dρ,K(r,0) = 0.(5)

For K(r,r′) we have a Goursat-type problem (equivalent to the Schr¨ odinger

equation):

LrK(r,r′) = Lr′0K(r,r′),

q(r) = −2

r

0 < r′≤ r,

d

dr

K(r,r)

r

,K(r,0) = 0.

Now, in analogy with the fixed–ℓ problem a Gel’fand-Levitan-Marchenko-

type integral equation is written up for K(r,r′):

K(r,r′) = g(r,r′) −

?r

0

dρρ−2K(r,ρ)g(ρ,r′),r ≥ r′. (6)

Since K(r,r′) is unique this integral equation must have a unique solution

for K(r,r′) to make sense.Furthermore, for consistency with the original

Schr¨ odinger equation, g(r,r′) must satisfy

Lr0g(r,r′) = Lr′0g(r,r′),g(r,0) = g(0,r′) = 0 (7)

and through the integral equation

q(r) = −2

r

d

dr

K(r,r)

r

(8)

3

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is maintained. g(r,r′) is a good candidate for approximation since its differen-

tial equation can be satisfied trivially. For instance if the angular momentum

expansion [5]

g(r,r′) =

?

is imposed then for the γℓ(r,r′) functions we have only the potential-independent

restrictions of

ℓ

cℓγℓ(r,r′),(9)

Lr0γℓ(r,r′) = Lr′0γℓ(r,r′),γℓ(r,0) = γℓ(0,r′) = 0,

∀ℓ.(10)

and the information is only stored in the cℓexpansion coefficients.

2Cox–Thompson (CT) method

In the framework of the method proposed by Cox and Thompson [3] one takes

the following separable form for the γℓ(r,r′) functions

γℓ(r,r′) = uℓ(min(r,r′))vℓ(max(r,r′)),(11)

which is also the Green’s function of the q ≡ 0 radial Schr¨ odinger equation for

the ℓth partial wave:

?d2

dr2+ 1 −ℓ(ℓ + 1)

r2

?

γℓ(r,r′) = δ(r − r′);(12)

and the summation in g(r,r′) runs only over a finite set S of ℓ’s:

g(r,r′) =

?

ℓ∈S

cℓuℓ(min(r,r′))vℓ(max(r,r′)),(13)

containing the Riccati-Bessel functions, connected to the Bessel and Neumann

functions by

uℓ(r) =

?πr

2Jℓ+1

2(r),vℓ(r) =

?πr

2Yℓ+1

2(r).(14)

For solving the GLM-type equation (6) we use the separable ansatz

K(r,r′) =

?

L∈T

AL(r)uL(r′)(15)

with a finite set T of ”shifted angular momenta” satisfying S ∩ T = ∅ and

|S| = |T|. The use of such an ansatz can be motivated by the following result

ascertained from [4].

Proposition 2.1. If r′−1/2K(r,r′) = O(1), r′→ 0 is imposed on K(r,r′) and

the L numbers are restricted to L > −0.5 then

cℓ=

?

L∈T(ℓ(ℓ + 1) − L(L + 1))

ℓ′∈S,ℓ′?=ℓ(ℓ(ℓ + 1) − ℓ′(ℓ′+ 1))

?

⇔

K(r,r′) =

?

L∈T

AL(r)uL(r′). (16)

Where {cℓ} ↔ {L} is a one-to-one mapping.

4

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Also, this form provides a reasonably easy way to solve the GLM-type inte-

gral equation. In fact it makes a system of algebraic equations instead of the

integral equation, namely

?

L∈T

AL(r)uL(r)v′

ℓ(r) − u′

L(r)vℓ(r)

ℓ(ℓ + 1) − L(L + 1)

= vℓ(r),ℓ ∈ S.(17)

The parameters of the set T can be obtained from the phase shifts {δℓ}ℓ∈S

through the transformation equation (5) which in terms of the L’s takes the

form

?

Taking both equation (18) and (17) for large r’s, i.e. at r → ∞ we get the

system of nonlinear equations [11, 12, 13] connecting {δℓ}ℓ∈Sto T:

ϕℓ(r) = uℓ(r) −

L∈T

AL(r)uL(r)u′

ℓ(r) − u′

L(r)uℓ(r)

ℓ(ℓ + 1) − L(L + 1)

(18)

e2iδℓ=1 + iK+

1 − iK−

ℓ

ℓ

,ℓ ∈ S,(19)

with

K±

ℓ=

?

L∈T

?

ℓ′∈S

[Msin]ℓL[M−1

cos]Lℓ′e±i(ℓ−ℓ′)π/2

ℓ ∈ S,(20)

?

Msin

Mcos

?

ℓL

=

1

L(L + 1) − ℓ(ℓ + 1)

?

sin((ℓ − L)π/2)

cos((ℓ − L)π/2)

?

,ℓ ∈ S, L ∈ T.

(21)

After some investigation of this nonlinear system of equations one can conclude

that for a given set of phase shifts it yields an infinity of solutions, each giving rise

to a potential. It was our result in [2] that one can select a physical potential

(perhaps uniquely) with the aid of a consistency check.

revisited in Sec. 4 and then it is applied for the special cases of one and two

dimensions, |S| = |T| = 1, 2.

This check is first

3Some new results concerning the CT potential

Before continuing with the consistency check it is worthwhile to further study

the CT method particularly concerning the potential one can obtain by applying

it. We address the case when only a finite number of input phase shifts are used

to construct the potential.

3.1 Asymptotics

First, we show that the potential is generally not compactly supported nor is of

long-range. To see this define the functions {Aa

L(r)}L∈T by the limit

AL(r) = Aa

L(r) + o(1),r → ∞,L ∈ T.(22)

5

Page 6

We infer from [13] that the asymptotic coefficients Aa

L(r) assume the form

Aa

L(r) = aLcosr + bLsinr (23)

where aL’s and bL’s are constant depending on all the elements of both S and

T:

?

?

This form in turn implies that

L∈T

aL

cos(π

2(ℓ − L))

L(L + 1) − ℓ(ℓ + 1)= cos

?

?

ℓπ

2

?

?

,ℓ ∈ S,(24)

L∈T

bL

cos(π

2(ℓ − L))

L(L + 1) − ℓ(ℓ + 1)= sin

ℓπ

2

,ℓ ∈ S.(25)

K(r,r) = αsin(2r) + β cos(2r) + γ + o(1),r → ∞

(26)

and thus

q(r) = 4β sin(2r) − αcos(2r)

r2

+ o(r−2),r → ∞

(27)

which means that the potential generally falls of like an inverse power of two.

One can give a necessary condition for the potential to decrease more rapidly

then O(r−2). One only needs

α = β = 0.(28)

The quantites α and β are given by

α =1

2

?

L∈T

?

aLcosLπ

2− bLsinLπ

2

?

(29)

β = −1

2

?

L∈T

?

aLsinLπ

2+ bLcosLπ

2

?

.(30)

Alternatively, K(r,r) can be written as a series in ℓ (this formula does not hold

for K(r,r′) with r ?= r′),

K(r,r) =

?

ℓ∈S

cℓϕℓ(r)vℓ(r), (31)

which can readily be seen from the GLM equation (6) taken at r = r′and the

CT formula (13) substituted for g(r,r′):

K(r,r) =

?

?

ℓ∈S

cℓuℓ(r)vℓ(r) −

?r

0

?r

dρρ−2K(r,ρ)

?

ℓ∈S

cℓuℓ(ρ)vℓ(r)(32)

=

ℓ∈S

cℓvℓ(r)

?

uℓ(r) −

0

K(r,ρ)uℓ(ρ)ρ−2dρ

?

,(33)

where the formula (5) for ϕℓ(r) has appeared. This allows for the alternative

conditions

?

ℓ∈S

(−1)ℓcℓBℓcosδℓ= 0, and

?

ℓ∈S

(−1)ℓcℓBℓsinδℓ= 0(34)

6

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involving the expansion coefficients, the input phase shifts and the normalization

constants of the partial wave functions.

These new results may serve as useful tools to check numerical results or

incorporated into a solution method they might provide a way to control the

undesirable oscillations of the inverse potential.

At this point we shortly discuss the applicability of the CT method. The

inverse potential is finite at the origin starting with a zero derivative (see e.g.

[15]) and possesses also a finite first moment?∞

(re)constructing inverse potentials. It was used to recover interaction potentials

(see e.g. [11, 12, 13]) for various physical system. Furthermore it is also possi-

ble to generalize the method to recover potentials having Coulomb tail e.g. by

replacing the Riccati-Bessel function by Coulomb wave functions [15, 16].

0rq(r)dr < ∞. At fixed scat-

tering energy the CT procedure is thus a particularly successful method for

3.2CT potentials regarding statistics of colliding particles

In real-life scattering experiments one encounters cases when the colliding par-

ticles are zero spin bosons. Then it is well-known that only the even numbered

partial waves contribute to the scattering amplitude because of the symmetry

of the wave function. In such cases with the CT method it is possible to use

only the experimentally available phase shift data corresponding to even partial

waves. It is then reasonable to ask for the value of phase shifts provided by the

CT potential for the odd partial waves. It turns out that these odd phase shifts

are zero.

To see this let us derive the phase shifts of the CT potential which can be

obtatined from Eq. (18) taken at r → ∞:

ϕj(r) = sin(r − jπ/2) −

?

L∈T

Aa

L(r)

sin((j − L)π/2)

j(j + 1) − L(L + 1)+ o(1),r → ∞. (35)

Now the trigonometric form (23) is substituted for Aa

cases that we consider:

i) the set S of input angular momenta consists of only even numbers,

ii) S consists of odd numbers.

In the first case we have bL= 0 ∀L ∈ T while in the second aL= 0 ∀L ∈ T.

This is deductible from Eqs. (24) and (25) which assume the forms

L(r). There are two special

?

?

L∈T

?

?

aL

bL

?

?

cos?Lπ

sin?Lπ

2

?

?

L(L + 1) − ℓ(ℓ + 1)=

?

?

1

0

?

?

,ℓ ∈ S (36)

L∈T

aL

bL

2

L(L + 1) − ℓ(ℓ + 1)=

0

1

,ℓ ∈ S(37)

for the cases i) and ii) respectively. One can assume cos?Lπ

2

??= 0 and sin?Lπ

2

??=

0, since the matrix M with elements MℓL= (L(L+1)−ℓ(ℓ+1))−1is invertible

7

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(it is a Cauchy matrix) and cannot be singular unless T ∩ S ?= ∅ which is ex-

cluded by assumption. We thus have bL= 0 ∀L ∈ T for i) and aL= 0 ∀L ∈ T

for ii). Now

?

(−1)j+1cosr,

which implies

sin(r − jπ/2) =

(−1)jsinr,j even

j odd

(38)

ϕj(r) = Bjsin(r−jπ/2)+o(1)for i) and odd j, or for ii) and even j. (39)

In other words the CT phase shifts of the opposite parities are exactly zero.

Notice that the CT method allows the construction of potentials which are

transparent for half the partial waves (being even or odd in parity).

Applications suggest that if dealing with input partial wave data of one parity

the performance of the CT method with the same number of input diminishes

compared to the case when data with both parity is employed. Therefore our

sum rules derived in the previous subsection can be extremely useful to improve

performance by supressing oscillations of the potential.

of our sum rules simplify (due to Bℓcosδℓ = 1, ℓ ∈ S in the even case and

Bℓsinδℓ= 1, ℓ ∈ S in the odd case [14]) to

Also note that one

?

ℓ∈S

cℓ= 0(40)

while the other becomes?

ℓ∈Scℓtanδℓ= 0 and?

ℓ∈Scℓcotδℓ= 0 for the even

and the odd case, respectively.

4Consistency check

4.1General condition

The kernel g(r,r′) only makes sense if the integral equation is uniquely solvable

with it. As Eq. (6) can be viewed as a Fredholm type integral equation of the

second kind for fixed r, viz. if rr′κ(r,r′) = K(r,r′) and rr′γ(r,r′) = g(r,r′) we

have

κ(r,r′) = γ(r,r′) −

?r

0

γ(r′,ρ)κ(r,ρ)dρ.(41)

According to Fredholm’s alternative the Fredholm determinant thereof must be

nonzero for all fixed r > 0. When using the CT ansatz the Fredholm determinant

of the integral equation becomes the determinant of the system of the algebraic

equations (17),

D(r) = det

??uL(r)v′

ℓ(r) − u′

L(r)vℓ(r)

ℓ(ℓ + 1) − L(L + 1)

?

ℓL

?

.(42)

It is easy to see that for X ∈ Ω = {x : D(x) = 0, x ∈ R+} one gets

lim

x→X

?x

0

tq(t)dt = ±∞,(43)

8

Page 9

thus the potential is not in L1,1= {q :

have [3]

?∞

One can conclude the following [2].

?∞

?

0t|q(t)|dt < ∞}. While for Ω = ∅ we

0

tq(t)dt =

?

L∈T

ℓ∈S(L − ℓ)

L?=L′∈T(L − L′)< ∞.

?

(44)

Theorem 4.1. If and only if D(r) ?= 0 on r > 0 we get a unique solution of the

GLM-type integral equation in C2(R+×R+) and from that an inverse potential

in L1,1= {q :

?∞

From [7] we know that D(r) ?= 0 on r > 0 is not the case for arbitrary choice

of S and T: e.g. there it was shown, that if S = {0} and L = {2} then D(r) = 0

at some r > 0.

With the help of Theorem 4.1 we can convince ourselves that a particular

CT inverse potential obtained numerically from arbitrary data is integrable or

not. If it is so then that potential will generate the input phase shifts.

Also, using the theorem it is possible to determine the admissible set of L

numbers for given ℓ’s. In the one-ℓ case we get a straightforward admissible

set of L’s, however in higher dimensions the formulation becomes extremely

involved.

0t|q(t)|dt < ∞}.

4.2One dimensional case

In this case – S = {ℓ}, |S| = 1 – the function W(uL,vℓ)(r) ≡ uL(r)v′

u′

L(r)vℓ(r) must be examined carefully. This was performed in [2] and thus it

shall not be discussed here in detail. We only give the key idea of the proof:

the Wronskian

uL(r)v′

ℓ(ℓ + 1) − L(L + 1)

ℓ(r) −

ℓ(r) − u′

L(r)vℓ(r)

(45)

is nonzero on R+if and only if the constituent Bessel functions (JL+1/2(x) and

Yℓ+1/2(x)) are interlaced. This is deductible from the observation

D(r) =uL(r)v′

ℓ(r) − u′

L(r)vℓ(r)

ℓ(ℓ + 1) − L(L + 1)

=

?r

0

uL(ρ)vℓ(ρ)ρ−2dρ.(46)

We could prove the next theorem of broader interest [10]

Theorem 4.2. The positive zeros of the Bessel functions Jν(x), J′

Y′

ν(x), Jν+ε(x), Yν+ε(x) for nonnegative orders, ν ≥ 0 are interlaced according

to the inequalities

ν(x), Yν(x),

ν ≤ j′

ν,1< yν,1< yν+ε,1< y′

ν,1< jν,1< jν+ε,1< j′

ν,s+1< ...(47)

if and only if 0 < ε ≤ 1 (otherwise yν+ε,s> jν,sfor some s and yν,s′+1> jν+ε,s′

for some s′).

The following gives the admissible set |ℓ − L| ≤ 1.

9

Page 10

Theorem 4.3. W(uL,vℓ)(r) has no roots on r ∈ R+, that is at N = 1 the

GLM-type equation is uniquely solvable for the CT method with S = {ℓ} and

T = {L} if and only if |L − ℓ| ≤ 1. ℓ ∈ (−0.5,∞), L ∈ (−0.5,∞) is supposed.

This result allows us to choose uniquely from the solutions of the system of

equations (19) as the solution for one input phase shift is

L = ℓ −2

πδℓ+ 2k,k ∈ Z, (48)

which for δℓ∈?−π

2,π

2

?in conjunction with the previous theorem yields

L = ℓ −2

πδℓ. (49)

At low energies it may happen that only one partial wave contributes mostly

to the scattering amplitude (e.g. for some partial wave resonances). It is worth-

while to look at the phase shifts yielded by the CT inverse potential at the

one-phase-shift level to get a sense of the quality of the inversion procedure.

Deductible from Eq. (18) is the following formula

tanδℓ=

?

0,ℓ odd

ℓ even

L(L+1)

L(L+1)−ℓ(ℓ+1)tanδ0,

(50)

which specifies the phases of the CT potential for S = {0}.

If e.g. the phase shift is restricted to describe an attractive potential (i. e.

δ0< 0) the bound

?

tanδℓ≤

0,ℓ odd

ℓ even

4

15ℓ2tanδ0,

(51)

can be found using Eq. (50). This result assures the proper reproduction of the

phase shifts by the CT potential.

To illustrate these results Fig. 1 shows synthetic test potentials correspond-

ing to ℓ = 0, δ0= 0.2π obtained by the CT method. In addition to the L1,1

potential an inconsistent one is also shown where k in Eq. (48) is chosen to be

other than zero. As indicated before we get a non-integrable potential.

4.3Two dimensional case

Already in this case the Fredholm determinant becomes complicated, which is

also apparent from it’s integral representation

D(r) =

?r

0

?r

0

uL1(ρ)uL2(ρ′)(vℓ1(ρ)vℓ2(ρ′) − vℓ1(ρ′)vℓ2(ρ))ρ−2ρ′−2dρdρ′. (52)

Therefore instead of the analytical treatment we determine the admissible set

of L’s numerically for some choices of ℓ’s.

10

Page 11

Figure 1: Potentials obtained by the CT method corresponding to ℓ = 0, δ0=

0.2π with L = −0.4 (solid line) and L = 1.6 (dashed line).

012345

r

6789 10

−2

0

2

4

6

8

10

q(r)

Integrable

Non−integrable

Our numerical method was to check the determinant at the points of a fine

lattice on the L1–L2 quarter plane of R+× R+whether it has any zeros on

(0,Λ), where Λ is a large number chosen to be great enough for

D(r > Λ) = const. + ε(r),

|ε(r)| < ε(53)

with a very small ε. This can be done since every D(r) in any dimensions

|S| < ∞ has only a finite number of zeros, since the constituent Wronskians all

tend to constants at large r distances, viz.

W(r) = uL(r)v′

ℓ(r) − u′

L(r)vℓ(r) = cos

?

(ℓ − L)π

2

?

+ O

?1

r

?

,r → ∞. (54)

On Fig. 2 the admissible sets of {L1,L2} pairs are depicted for the particular

choices S1= {1,3} and S2= {1,2}.

In the next example (Fig. 3) we calculated some possible {L1,L2} pairs for

a given {δℓ1,δℓ2}. We note that only one of them is inside the permitted domain

and could only find a single solution of the system of non-linear equation that

is permitted by the consistency condition. Again the L1,1and an inconsistent

potential is shown.

5Conclusion

After recalling some crucial results of the inverse scattering problem for the

Schr¨ odinger equation at fixed scattering energy we reviewed a particular con-

structive scheme, the Cox-Thompson (CT) method for recovering scattering

potentials responsible for a finite set of given phase shifts.

11

Page 12

Figure 2: The admissible sets of the T elements for (a) S1 = {1,3} and (b)

S2= {1,2} denoted by blank areas.

(a)

(b)

Figure 3:

{0,1}, with phase shifts calculated from a Woods-Saxon potential.

{−0.3056,0.9295} (solid line) and T = {1.0650,1.7016} (dashed line). Also,

the original potential, q(r) = −?1 + e2.5·(r−1)?−1, yielding the phase shifts

Potentials obtained by the CT method corresponding to S =

T =

(δ0= 0.4389, δ1= 0.1246) is depicted (dotted line).

012345

r

678910

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

q(r)

Integrable

Non−integrable

Original

12

Page 13

We have analyzed the asymptotic properties of the CT potentials and from

that we have obtained a sum rules for the expansion coefficients (Eq. (28,34)).

Then we considered CT potentials corresponding to specific quantum statistical

characters of the colliding pattern. If the scattered particles are zero spin bosons

(e.g. oxygen atoms or carbon nuclei) the CT potentials are constructed from

even phase shifts being accessible for measurements. We have shown that in this

case the CT potentials in addition to reproducing the input even phase shifts

give exactly zero phases for the odd partial waves.

Then we turned to the consistency check. It has been shown that for the

potential to be integrable (or more precisely, to belong to the function class L1,1)

a condition must be fulfilled for the intermediate quantities L of shifted angular

momenta of the CT method. We discussed the case when only one input phase

shift is used in which circumstance we have explicit uniqueness. The two-phase-

shift case has also been discussed. The condition for the L numbers to produce

an integrable potentials has been obtained through a numerical calculation for

two particular cases and the results have been illustrated in Fig.3.

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