# An algebraic form of the Marchenko inversion.
Partial waves with orbital momentum $l\geq 0$

N. A. Khokhlov<EMAIL_ADDRESS>Southwest State University, Kursk,
Russia

###### Abstract

We present a generalization of the algebraic method for solving the Marchenko
equation (fixed-$l$ inversion) for any values of the orbital angular momentum
$l$. We expand the Marchenko equation kernel in a separable form using a
triangular wave set. The separable kernel allows a reduction of the equation
to a system of linear equations. We obtained a linear expression of the kernel
expansion coefficients in terms of the Fourier series coefficients of
$q(1-S(q))$ function ($S(q)$ is the scattering matrix) depending on the
momentum $q$. The linear expression is valid for any orbital angular momentum
$l$. The kernel expansion coefficients are determined by the scattering data
in the finite range $0\leq q\leq\pi/h$. In turn, the thus defined Marchenko
kernel of the equation allows one to find the potential function of the radial
Schrödinger equation with $h$-step accuracy.

quantum scattering, inverse problem, Marchenko theory, algebraic method,
numerical solution

###### pacs:

24.10.Ht, 13.75.Cs, 13.75.Gx

##  I INTRODUCTION

The inverse problem (IP) of quantum scattering is essential for many physical
applications. One of the most important such applications is the interparticle
potential extraction from scattering data. The fixed-$l$ IP considered here is
usually solved within the framework of Marchenko, Krein, and Gelfand-Levitan
theories Gelfand ; Agranovich1963 ; Marchenko1977 ; Krein1955 ; Levitan1984 ;
Newton ; Chadan . Development of accurate and unambiguous methods for solving
this problem remains a fundamental challenge Sparenberg1997 ; Sparenberg2004 ;
Kukulin2004 ; Pupasov2011 ; Mack2012 . The ill-posedness of the IP
significantly complicates its numerical solution.

This paper considers a new algebraic method for solving the fixed-$l$ inverse
problem of quantum scattering theory. We derive the method from the Marchenko
theory. Marchenko theory was successfully applied by H.V. von Geramb and H.
Kohlhoff to recover nucleon-nucleon partial potentials from partial-wave
analysis (PWA) data up to the inelastic threshold
($E_{\text{lab}}\approx~{}280$ MeV) Geramb1994 ; Kohlhoff1994 . They used
rational fraction expansions of partial $S$-matrices. This expansion allows
one to obtain an analytical solution to the Marchenko equation (Bargman-type
potentials). Optical model nucleon-nucleon partial potentials were recovered
from PWA data up to 3 GeV using a similar approach Khokhlov2006 ; Khokhlov2007
. It is not clear whether such a procedure converges with an increase in the
$S$-matrix approximation accuracy. We approximate the integral kernel by a
separable series in the triangular single wave set. Thus, the Marchenko
equation is solved analytically as in Refs. Geramb1994 ; Kohlhoff1994 ;
Khokhlov2006 ; Khokhlov2007 . The expansion coefficients of the integral
kernel are obtained from the Fourier series coefficients of the function
$q\left(1-S(q)\right)$ on a finite range ($0\leq q\leq\pi/h$) of the momentum
$q$. The $h$ value determines a required accuracy of the potential function.

##  II Marchenko equation in an algebraic form

The radial Schrödinger equation is

$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0.$
(1)

The Marchenko equation Agranovich1963 ; Marchenko1977  is a Fredholm integral
equation of the second kind:

$F(x,y)+L(x,y)+\int_{x}^{+\infty}L(x,t)F(t,y)dt=0$ (2)

The kernel function is defined by the following expression

$F(x,y)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}h_{l}^{+}(qx)[1-S(q)]h_{l}^{+}(qy)dq\\\
+\sum_{j=1}^{n_{b}}h_{l}^{+}\left(\tilde{q}_{j}x\right)M_{j}^{2}h_{l}^{+}\left(\tilde{q}_{j}y\right)\\\
=\frac{1}{2\pi}\int_{-\infty}^{+\infty}h_{l}^{+}(qx)Y(q)h_{l}^{+}(qy)dq$ (3)

where $h_{l}^{+}(z)$ is the Riccati-Hankel function, and

$Y(q)=\left[1-S(q)-i\sum_{j=1}^{n_{b}}M_{j}^{2}\left(q-\tilde{q}_{j}\right)^{-1}\right],$
(4)

Experimental data entering the kernel are

$\left\\{S(q),(0<q<\infty),\tilde{q}_{j},M_{j},j=1,\ldots,n\right\\},$ (5)

where $S(q)=e^{2\imath\delta(q)}$ is a scattering matrix dependent on the
momentum $q$. The $S$-matrix defines asymptotic behavior at
$r\rightarrow+\infty$ of regular at $r=0$ solutions of Eq. (1) for $q\geq 0;\
\tilde{q}_{j}^{2}=E_{j}\leq 0,E_{j}$ is $j$-th bound state energy
($-\imath\tilde{q}_{j}\geq 0$); $M_{j}$ is $j$-th bound state asymptotic
constant.

The potential function of Eq. (1) is obtained from the solution of Eq. 2

$V(r)=-2\frac{dL(r,r)}{dr}$ (6)

Many methods for solving Fredholm integral equations use series expansion of
the equation kernel. eprint7 ; eprint8 ; eprint9 ; eprint10 ; eprint11 ;
eprint12 ; eprint13 ; eprint14 . We also use this approach.

We introduce auxiliary functions:

$F_{m}(z)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{e^{\imath
qz}Y(q)dq}{q^{m}},$ (7)

then

$\frac{d^{k}F_{m}(z)}{dz^{k}}=\imath^{k}F_{m-k}(z),\ (k=1,2,\dots,m).$ (8)

We use transformations

$\hat{K}_{z,l}f(z)=z^{l+1}\left(-\frac{1}{z}\frac{d}{dz}\right)^{l}\left[z^{-1}f(z)\right]\\\
\equiv(-1)^{l}{\sum_{n=0}^{l}\frac{(2l-n)!}{n!(l-n)!}(-2z)^{n-l}\frac{d^{n}f(z)}{dz^{n}}}.$
(9)

Thus (Abramowitz , Eqs. 10.1.23-10.1.26)

$\hat{K}_{z,l}e^{\pm\imath qz}=q^{l}h^{\pm}_{l}(qz).$ (10)

and

$\hat{K}_{y,l}\hat{K}_{x,l}F_{2l}(x+y)={\sum_{n_{1},n_{2}=0}^{l}\frac{(2l-n_{1})!}{n_{1}!(l-n_{1})!}\frac{(2l-n_{2})!}{n_{2}!(l-n_{2})!}(-2x)^{n_{1}-l}(-2y)^{n_{2}-l}\imath^{n_{1}+n_{2}}F_{2l-n_{1}-n_{2}}(x+y)}\\\
=\frac{1}{2\pi}\int_{-\infty}^{+\infty}h_{l}^{+}(qx)Y(q)h_{l}^{+}(qy)dq\equiv
F(x,y).$ (11)

Assuming the finite range $R$ of the bounded potential function, we
approximate $F_{m}(x+y)$ as follows:

$\displaystyle F_{m}(x+y)\approx\sum_{k=-2N}^{2N}f_{m,k}H_{k}(x+y)$ (12)
$\displaystyle\approx\sum_{k,j=0}^{N}\Delta_{k}(x)f_{m,k+j}\Delta_{j}(y)$ (13)

where $f_{m,k}\equiv F_{m}(kh)$, $h$ is some step, and $R=Nh$. The used basis
sets are

$\left.\begin{array}[]{l}H_{0}(x)=\left\\{\begin{array}[]{lr}1&\text{if }0\leq
x\leq h,\\\ 0&\text{otherwise,}\end{array}\right.\\\
H_{n}(x)=H_{0}(x-hn).\end{array}\right\\}$ (14)
$\left.\begin{array}[]{l}\Delta_{0}(x)=\left\\{\begin{array}[]{lr}1-|x-0.25|/h&\text{if
}|x-0.25|\leq h,\\\ 0&\text{otherwise,}\end{array}\right.\\\
\Delta_{n}(x)=\Delta_{0}(x-hn).\end{array}\right\\}$ (15)

We use bases set $\Delta_{i}(x)\Delta_{j}(y)$ shifted by vector
$(0.25h,0.25h)$ compared to the set used previously MyAlg1 ; MyAlg2 . The
basis sets are illustrated in the (Fig. 1).

Figure 1: The basis set $H_{n}\equiv H_{n}(x+y)$ (Eq. (14) is shown as
trapezoid (triangle for $n=0$) domains where $H_{n}(x+y)=1$, and elsewhere
$H_{n}(x+y)=0$. The domains are bounded by lines $x=0$, $y=0$, and
$x+y=h(n-1)$. The basis set $\Delta_{i}(x)\Delta_{j}(y)$ (Eq. (15)) is shown
as projections (points) of the corresponding regular square pyramids apexes on
the $xy$-plane. $\Delta_{i}(x)\Delta_{j}(y)=1$ at $x=(0.25+i)h,\ y=(0.25+j)h$
(apex of the $ij$-pyramid). The pyramids bases are $(2h\times 2h)$ squares on
the $xy$-plain with sides parallel to the $x$ and $y$ axes. On sides of the
corresponding square (as well as outside them) $\Delta_{i}(x)\Delta_{j}(y)=0$.

Decreasing the step $h$, one can approach $F_{m}(x+y)$ arbitrarily close at
all points with both sets. Coefficients $f_{m,k}$ are same for both
approximations.

The Fourier transform of the basis set Eq. 12

$\tilde{\mathrm{H}}_{k}(q)=\int_{-\infty}^{\infty}\mathrm{H}_{k}(x)e^{-\imath
qx}dx=\frac{\imath\left(e^{-\imath qh}-1\right)}{qe^{\imath qhk}}.$ (16)

The Fourier transform of Eq. (7) yields

$\frac{Y(q)}{q^{m}}\approx\sum_{k=-2N}^{2N}f_{m,k}\tilde{\mathrm{H}}_{k}(q)=\sum_{k=-2N}^{2N}f_{m,k}\frac{\imath\left(e^{-\imath
qh}-1\right)}{qe^{\imath qhk}}.$ (17)

We rearrange the last relationship

$Y(q)/q^{m-1}=\imath\sum_{k=-2N}^{2N}f_{m,k}\left(e^{-\imath
qh}-1\right)e^{-\imath qhk}\\\
=\imath\sum_{k=-2N+1}^{2N}\left(f_{m,k-1}-f_{m,k}\right)e^{-\imath
qhk}+\imath\left(-f_{m,-2N}\right)e^{\imath qh2N}\\\
+\imath\left(f_{m,2N}\right)e^{-\imath qh(2N+1)}.$ (18)

Thus, the left side of the expression is represented as a Fourier series on
the interval $-\pi/h\leq q\leq\pi/h$.

$f_{m,k-1}-f_{m,k}=-\frac{\imath
h}{2\pi}\int_{-\pi/h}^{\pi/h}Y(q)\frac{e^{\imath qhk}dq}{q^{m-1}}$ (19)

for $k=-2N,\ldots,2N$. We solve the system (19) recursively from $k=2N+1$
($f_{m,2N+1}=0$) for fixed $m$:

$f_{m,k}=-\frac{\imath h}{2\pi}\int_{-\pi/h}^{\pi/h}\frac{e^{\imath
qh(k+1)}\left(1-e^{\imath qh(2N-k+1)}\right)}{\left(1-e^{\imath
qh}\right){q^{m-1}}}Y(q)dq.$ (20)

The $F(x,y)$ is defined by $f_{m,k}$ $(m=0,1,\dots,2l)$, $k=0,1,\dots,2N$ from
Eqs. (11), and (13) as

$F(x,y)\approx\sum_{k,j=0}^{N}\Delta_{k}(x)F_{k,j}\Delta_{j}(y),$ (21)

where

$\displaystyle
F_{k,j}={{\sum_{n_{1},n_{2}=0}^{l}\frac{(2l-n_{1})!}{n_{1}!(l-n_{1})!}\frac{(2l-n_{2})!}{n_{2}!(l-n_{2})!}(-2(k+0.25)h)^{n_{1}-l}(-2(j+0.25)h)^{n_{2}-l}\imath^{n_{1}+n_{2}}f_{2l-n_{1}-n_{2},k+j}}}$
(22) $\displaystyle=-\frac{\imath
h}{2\pi}\int_{-\pi/h}^{\pi/h}h_{l}^{+}(qkh)\frac{e^{\imath
qh}\left(1-e^{\imath qh(2N-k-j+1)}\right)}{1-e^{\imath
qh}}Y(q)h_{l}^{+}(qjh)qdq.$ (23)

Thus, the range of known scattering data defines value of $h$ and, therefore,
the inversion accuracy.

We solve Eq. (2) substituting

$L(x,y)\approx\sum_{j=0}^{N}P_{j}(x)\Delta_{j}(y)$ (24)

Substitution of Eqs. (21) and (24) into Eq. (2), and linear independence of
the basis functions give

$\sum_{m=0}^{N}\left(\delta_{j\,m}+\sum_{n=0}^{N}\left[\int_{x}^{max((m+0.25)h,(n+0.25)h)}\Delta_{m}(t)\Delta_{n}(t)dt\right]F_{n,j}\right)P_{m}(x)=-\sum_{k=0}^{N}\Delta_{k}(x)F_{k,j}$
(25)

We define

$\zeta_{n\,m\,p}=\int_{(p+0.25)h}^{max((m+0.25)h,(n+0.25)h)}\Delta_{m}(t)\Delta_{n}(t)dt=\frac{h}{6}\left(2\delta_{n\,m}\left(\delta_{n\,p}+2\eta_{n\geq
p+1}\right)+\delta_{n\,(m-1)}\eta_{n\geq p}+\delta_{n\,(m+1)}\eta_{m\geq
p}\right),$ (26)

where $\delta_{k\,p}$ are the Kronecker symbols $\delta_{k\,p}$, and

$\eta_{a}=\left\\{\begin{array}[]{lr}1&\text{if }a\text{ is true},\\\
0&\text{otherwise,}\end{array}\right.$ (27)

Since $\Delta_{k}(hp)\equiv\delta_{k\,p}$, we finally get a system of
equations

$\sum_{m=0}^{N}\left(\delta_{j\,m}+\sum_{n=p}^{N}\zeta_{n\,m\,p}F_{n,j}\right)P_{p,m}=-F_{p,j},$
(28)

for $P_{k}(h(p+0.25))\equiv P_{p,k}$ $(p,k=0,\dots,N)$ ($j,p=0,\dots,N$).

Solution of Eq. (28) gives $P_{p,k}$. We calculate potential values at points
$r=hp$ $(p=0,\dots,N)$ from Eq. (6) by some finite difference formula.

##  III Results and Conclusions

We tested the developed approach by restoring the potential function
$V(r)=-3\exp(-3r/2)$ from the corresponding scattering data. Results are
presented in Figs. 2, 3, where $h=0.04$, $R=4$.

Figure 2: Data used to reconstruct $V(r)=V_{0}\exp(-ar)$, where $V_{0}=-3\
fm^{-2}=-124.5\ MeV$, $a=1.5\ fm^{-1}$. Units correspond to the $NN$ system.

Figure 3: Initial and reconstructed potentials: $V(r)=V_{0}\exp(-ar)$, where
$V_{0}=-3\ fm^{-2}=-124.5\ MeV$, $a=1.5\ fm^{-1}$. Units correspond to the
$NN$ system.

The input $S$-matrix was calculated at points shown in Fig. 2 up to $q=8$. The
$S$-matrix was interpolated by a quadratic spline in the range $0<q<8$. The
$S$-matrix was approximated as asymptotic $S(q)\approx\exp(-2i\alpha/q)$ for
$q>8$, where $\alpha$ was calculated at $q=8$.

Thus, we presented a general solution of the quantum scattering inverse
problem for the any orbital angular momentum $l$. The algorithm of the
solution is as follows. We set the step value $h$, which determines a required
accuracy of the potential. From the experimental data, we determine $F_{k,j}$
using Eqs. (23). Solution of Eqs. (28) gives values of $P_{k}(hp)$
($p=0,\dots,N$). The values of the potential function (6) are determined by
some finite difference formula. Expressions (21-28) give a method for the
Marchenko equation’s numerical solution for an arbitrary orbital angular
momentum $l$.

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