Appendix A
SOLUTIONS OF THE NONLOCAL EQUATION
The inhomogeneous Schrödinger wave equation with a symmetric
nonlocal potential \(V_{\ell}(r,s)\) for all partial waves can be
written as
$$\left({\frac{d^{2}}{dr^{2}}+k^{2}+\frac{\ell\left({\ell+1)}\right)}{r^{2}}}\right)f_{\ell}(k,r)$$
(A.1)
$${}=\int\limits_{0}^{\infty}{V_{\ell}(r,s)f_{\ell}(k,s)ds},$$
where \(V_{\ell}(r,s)\) is the rank-1 Graz separable potential and is
expressed as
$$V_{\ell}(r,s)=\lambda_{\ell}g_{\ell}(r)g_{\ell}(s)$$
(A.2)
with the form factor of the potential
$$g_{\ell}(r)=2^{-\ell}\left({\ell!}\right)^{-1}r^{\ell}e^{-\beta_{\ell}r}.$$
(A.3)
The irregular solution \(f_{\ell+}(k,r)\) to Eq. (A.1) can be
written in the terms of free particle irregular solution \(f_{\ell 0}(k,r)\) and irregular Green’s function \(G_{\ell}^{(I)}(r,s)\) as
$$f_{\ell+}(k,r)=f_{\ell 0}(k,r)+\lambda_{\ell}d_{\ell}(k)I_{\ell}(r,\beta_{\ell}),$$
(A.4)
where
$$I_{\ell}(r,\beta_{\ell})=\int\limits_{r}^{\infty}{g_{\ell}(s)G_{\ell}^{(I)}(r,s)ds}.$$
(A.5)
The quantities \(f_{\ell 0}(k,r)\), \(d_{\ell}(k)\) and the
irregular Green’s function \(G_{\ell}^{(I)}(r,s)\) [27] are
expressed as
$$f_{\ell 0}(k,r)=-i\left({2kr}\right)^{\ell+1}e^{i\left({kr-\frac{\ell\pi}{2}}\right)}$$
(A.6)
$${}\times\Psi(\ell+1,2\ell+2;-2ikr),$$
$$d_{\ell}(k)=\int\limits_{0}^{\infty}{g_{\ell}(s)f_{\ell+}(k,s)ds}$$
(A.7)
and
$$G_{\ell}^{(I)}(r,s)=\begin{cases}G_{\ell}^{(I)}(r,s),\quad s>r\\ 0,\quad s<r.\end{cases}$$
(A.8)
Multiplying Eq. (A.4) by \(g_{\ell}(r)\) and integrating over whole space
one obtains
$$d_{\ell}(k)=\int\limits_{0}^{\infty}{g_{\ell}(r)f_{\ell 0}(k,r)dr/D_{\ell}(k)},$$
(A.9)
where the Fredholm determinant \(D_{\ell}(k)\) associated with irregular
boundary condition reads as
$${}-\lambda_{\ell}\int\limits_{0}^{\infty}{\int\limits_{r}^{\infty}{g_{\ell}(r)g_{\ell}(s)G_{\ell}^{(I)}(r,s)dsdr}}.$$
To have an expression for \(f_{\ell+}(k,r)\) one needs to solve the integrals
involved in Eqs. (A.5), (A.9), and (A.10). To solve the indefinite integral
in Eq. (A.5) we proceed as follows. For all partial waves \(G_{\ell}^{(I)}(r,s)\) can be expressed in terms of confluent hypergeometric functions [7,
12] as
$$G_{\ell}^{(I)}(r,s)=i\left({2k}\right)^{2\ell+1}\frac{\Gamma(\ell+1)}{\Gamma(2\ell+2)}$$
(A.11)
$${}\times e^{-i\ell\pi}r^{\ell+1}s^{\ell+1}e^{ikr}e^{iks}$$
$${}\times\Big{\{}\!{\Phi(\ell+1,2\ell+2;-2ikr)\Psi(\ell+1,2\ell+2;-2iks)}$$
$${}-{\Phi(\ell+1,2\ell+2;-2iks)\Psi(\ell+1,2\ell+2;-2ikr)}\!\Big{\}}.$$
Substituting Eq. (A.11) in Eq. (A.5) together with Eq. (A.3) we have
$$I_{\ell}(r,\beta_{\ell})=i(2k)^{2l+1}r^{l+1}e^{ikr}\frac{\Gamma(\ell+1)}{\Gamma(2\ell+2)}$$
(A.12)
$${}\times e^{-i\ell\pi}2^{-\ell}\left({\ell!}\right)^{-1}\Biggl{[}\int\limits_{0}^{\infty}{s^{2\ell+1}e{}^{-\left({\beta_{\ell}-ik}\right)s}}$$
$${}\times\Big{\{}\Phi(\ell+1,2\ell+2;-2ikr)$$
$${}\times\Psi(\ell+1,2\ell+2;-2iks)$$
$${}-\Phi(\ell+1,2\ell+2;-2iks)$$
$${}\times\Psi(\ell+1,2\ell+2;-2ikr)\Big{\}}ds$$
$${}-\int\limits_{0}^{r}s^{2\ell+1}e{}^{-\left({\beta_{\ell}-ik}\right)s}\Big{\{}\Phi(\ell+1,2\ell+2;-2ikr)$$
$${}\times\Psi(\ell+1,2\ell+2;-2iks)$$
$${}-\Phi(\ell+1,2\ell+2;-2iks)$$
$${}\times\Psi(\ell+1,2\ell+2;-2ikr)\Big{\}}ds\Biggl{]}.$$
With the following standard integrals and relation [28–33]
$$\int\limits_{0}^{\infty}{e^{-\lambda z}}z^{\upsilon}\Phi(a,c;pz)dz$$
(A.13)
$${}=\frac{\Gamma(\upsilon+1)}{\lambda^{\upsilon+1}}{}_{2}F_{1}\left({a,\upsilon+1;c;\frac{p}{\lambda}}\right),$$
$$\int\limits_{0}^{\infty}{e^{-ax}}x^{s-1}\Psi(b,d;\mu x)dx$$
(A.14)
$${}=\frac{\Gamma(1+s-d)\Gamma(s)}{a^{s}\Gamma(1+b+s-d)}$$
$${}\times{}_{2}F_{1}\left({b,s;1+b+s-d;1-\frac{\mu}{a}}\right)$$
$$(\mathrm{Re}\ s>0,\;1+\mathrm{Re}\ s>\mathrm{Re}\ d),$$
$$\theta_{\sigma}(a,c;z)=\frac{1}{c-1}\Biggl{[}\Phi(a,c;z)$$
(A.15)
$${}\times\int{e^{-z^{\prime}}}(z^{\prime})^{\sigma+c-2}\bar{\Phi}(a,c;z^{\prime})dz^{\prime}$$
$${}-\bar{\Phi}(a,c;z)\int{e^{-z^{\prime}}}(z^{\prime})^{\sigma+c-2}\Phi(a,c;z^{\prime})dz^{\prime}\Biggl{]},$$
$${}_{2}F_{1}(a,b;b;z)=(1-z)^{-a}$$
(A.16)
and
$$\Psi(a,b;z)=\frac{\Gamma(1-b)}{\Gamma(a-b+1)}\Phi(a,b;z)$$
(A.17)
$${}+\frac{\Gamma(b-1)}{\Gamma(a)}\bar{\Phi}(a,b;z).$$
Equation (A.12) yields
$$I_{\ell}(r,\beta_{\ell})=i\left({2k}\right)^{2\ell+1}r^{l+1}e^{ikr}e^{-i\ell\pi}2^{-\ell}\left({\ell!}\right)^{-1}$$
(A.18)
$${}\times\Biggl{[}\frac{1}{\left({\beta_{\ell}-ik}\right)^{2\ell+2}(\ell+1)}$$
$${}\times{}_{2}F_{1}\left({\ell+1,2\ell+2;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)$$
$${}\times\Phi(\ell+1,2\ell+2;-2ikr)$$
$${}-\frac{\Gamma(\ell+1)}{\left({\beta_{\ell}-ik}\right)^{2\ell+2}}\left({\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)^{-\ell-1}$$
$${}\times\Psi(\ell+1;2\ell+2;-2ikr)$$
$${}-\left({\frac{1}{2ik}}\right)^{2\ell+2}\sum\limits_{n=0}^{\infty}{\left({\frac{\beta_{\ell}+ik}{2ik}}\right)}^{n}\left({\frac{1}{n!}}\right)$$
$${}\times\theta_{n+1}(\ell+1,2\ell+2;-2ikr)\Biggl{]}.$$
Equation (A.18) represents the single transform of the irregular Green’s
function with the form factor of the separable potential. To calculate the
Fredholm determinant \(D_{\ell}(k)\) we substitute Eq. (A.18) in Eq. (A.10),
use the standard integrals in Eqs. (A.13) and (A.14) along with [28–31]
$$\int\limits_{0}^{\infty}{e^{-bz}z^{c-1}\theta_{\sigma}(a,c;pz)}dz$$
(A.19)
$${}=\frac{\Gamma(\sigma+c-1)p^{\sigma}}{\sigma b^{\sigma+c}}{}_{2}F_{1}\left({1,\sigma+a;\sigma+1;\frac{p}{b}}\right),$$
$${}_{2}F_{1}(a,b;c;z)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$$
(A.20)
$${}\times{}_{2}F_{1}(a,b;a+b-c+1;1-z)$$
$${}+\left({1-z}\right)^{c-a-b}\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}$$
$${}\times{}_{2}F_{1}(c-a,c-b;c-a-b+1;1-z),$$
$${}_{2}F_{1}(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}$$
(A.21)
$${}\times\sum\limits_{n=0}^{\infty}{\frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}}\frac{z^{n}}{n!}$$
and
$${}_{2}F_{1}(a,b;c;z)$$
(A.22)
$${}=\left({1-z}\right)^{-a}{}_{2}F_{1}\left({a,c-b;c;\frac{z}{z-1}}\right)$$
to get
$$D_{\ell}(k)=1-\left({-1}\right)^{\ell+1}\lambda_{\ell}2^{-2\ell}e^{-i\ell\pi}\left({\ell!}\right)^{-2}$$
(A.23)
$${}\times\Biggl{[}\frac{1}{\left({\beta_{\ell}-ik}\right)^{2\ell+3}}\sum\limits_{n=0}^{\infty}{\frac{\left({-1}\right)^{n}}{n!}\left({\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)}^{n}$$
$${}\times\frac{\Gamma(n+2\ell+2)}{\left({\ell+1}\right)}{}_{2}F_{1}\!\!\left({1,n+\ell+2;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)$$
$${}+\frac{\Gamma(2\ell+2)}{\left({\ell+1}\right)\left({\beta_{\ell}^{2}+k^{2}}\right)^{\ell+1}}$$
$${}\times\left({\frac{-1}{\beta_{\ell}-ik}}\right){}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)\Biggl{]}.$$
To have a compact expression by removing the infinite
sum in the above equation we further use the integral
representation of the Gaussian hypergeometric function and
transformation relation [28-30]
$${}_{2}F_{1}(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}$$
(A.24)
$${}\times\int\limits_{0}^{1}{dtt^{b-1}\left({1-t}\right)^{c-b-1}\left({1-tz}\right)^{-a}}$$
and
$${}_{2}F_{1}(a,b;c;z)$$
(A.25)
$${}=\left({1-z}\right)^{c-a-b}{}_{2}F_{1}(c-a,c-b;c;z)$$
to have
$$D_{\ell}(k)=1-\frac{\lambda_{\ell}2^{-2\ell}\left({\ell!}\right)^{-2}\Gamma(2\ell+2)}{(\ell+1)\left({\beta_{\ell}-ik}\right)}$$
(A.26)
$${}\times\Biggl{[}\left({\beta_{\ell}^{2}+k^{2}}\right)^{-\ell-1}{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)$$
$${}-\frac{\left({2\beta_{\ell}}\right)^{-2\ell-1}}{\left({\beta_{\ell}-ik}\right)}{}_{2}F_{1}\left({1,-\ell;\ell+2;\left({\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)^{2}}\right)\Biggl{]}.$$
The above equation gives the desired expression for the
Fredholm determinant associated with the irregular boundary
condition.
Equation (A.9) in conjunction with Eqs. (A.6), (A.14), (A.25), and
(A.26) yields
$$d_{\ell}(k)=\frac{-ie^{-i\frac{\ell\pi}{2}}2^{-\ell}\left({\ell!}\right)^{-1}\left({2k}\right)^{\ell+1}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)^{2\ell+2}}$$
(A.27)
$${}\times\frac{\Gamma(2\ell+2)}{\Gamma(\ell+2)}\left({\frac{\beta_{\ell}-ik}{-2ik}}\right)^{2\ell+1}$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right).$$
Finally, Eq. (A.4) together with Eqs. (A.6), (A.18),
(A.26), and (A.27) reproduce the irregular solution for motion in
Graz separable potential that reads as
$$f_{\ell+}(k,r)=-i\left({2kr}\right)^{\ell+1}e^{i\left({kr-\frac{\ell\pi}{2}}\right)}$$
(A.28)
$${}\times\Psi(\ell+1,2\ell+2;-2ikr)$$
$${}-\frac{\lambda_{\ell}\Gamma(2\ell+2)r^{\ell+1}e^{i\left({kr+\frac{\pi}{2}}\right)}e^{-i\frac{\ell\pi}{2}}}{D_{\ell}(k)\Gamma(\ell+2)\left({\beta_{\ell}-ik}\right)2^{\ell-1}\left({\ell!}\right)^{2}}$$
$${}\times{}_{2}F_{1}(1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik})$$
$${}\times\Biggl{[}\frac{k^{\ell+1}\Gamma(\ell+1)}{\left({\beta_{\ell}^{2}+k^{2}}\right)^{\ell+1}}\Psi(\ell+1;2\ell+2;-2ikr)$$
$${}-\frac{e^{-i\ell\pi}}{2^{2\ell+2}k^{\ell+1}}\Biggl{\{}\frac{2ik}{\left({\ell+1}\right)\left({\beta_{\ell}-ik}\right)}$$
$${}\times{}_{2}F_{1}\left(1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}\right)$$
$${}\times\Phi(\ell+1;2\ell+2;-2ikr)$$
$${}+\sum\limits_{n=0}^{\infty}\left({\frac{\beta_{\ell}+ik}{2ik}}\right)^{n}\frac{1}{\left({n!}\right)}$$
$${}\times\Theta_{n+1}(\ell+1,2\ell+2;-2ikr)\Biggl{\}}\Biggl{]}.$$
Utilizing Eq. (A.28) along with the following formulae [31-33]
$$\frac{d^{n}}{dz^{n}}\Phi(a,c;z)=\frac{(a)_{n}}{(c)_{n}}\Phi(a+n,c+n;z),$$
(A.29)
$$\frac{d^{n}}{dz^{n}}\Psi(a,c;z)=\left({-1}\right)^{n}(a)_{n}$$
(A.30)
$${}\times\Psi(a+n,c+n;z)$$
and
$$\frac{d}{dz}\Theta_{\sigma}(a,c;z)=\left({\sigma-1}\right)$$
(A.31)
$${}\times\Theta_{\sigma-1}(a+1,c+1;z).$$
Equation (2) results
$$J_{\ell}(k,r)=k^{-1}\textrm{Im}\biggl{[}\left({2k}\right)^{2\ell+2}r^{2\ell+1}$$
(A.32)
$${}\times\left({ikr+\ell+1}\right)A_{1}+2ik\left({\ell+1}\right)\left({2kr}\right)^{2\ell+2}A$$
$${}+\frac{\lambda_{\ell}\Gamma(2\ell+2)\left({2k}\right)^{\ell+1}r^{2\ell+1}}{\Gamma(\ell+2)\left({\ell!}\right)^{2}2^{\ell-1}}\left({ikr+\ell+1}\right)$$
$${}\times\left\{{\frac{MX^{\ast}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}+\frac{M^{\ast}X}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}}\right\}$$
$${}+\frac{\lambda_{\ell}\Gamma(2\ell+2)\left({2k}\right)^{\ell+1}r^{2\ell+2}}{\Gamma(\ell+2)\left({\ell!}\right)^{2}2^{\ell-1}}$$
$${}\times\left\{{\frac{M_{1}X^{\ast}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}+\frac{2ik\left({\ell+1}\right)M^{\ast}Y}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}}\right\}$$
$${}+Pr^{2\ell+1}\left\{{\left({\ell+1+ikr}\right)MM^{\ast}+rM^{\ast}M_{1}}\right\}\biggl{]},$$
where
$$A=\Psi(\ell+1;2\ell+2;2ikr)$$
(A.33)
$${}\times\Psi(\ell+2;2\ell+3;-2ikr),$$
$$A_{1}=\Psi(\ell+1;2\ell+2;-2ikr)$$
(A.34)
$${}\times\Psi(\ell+1;2\ell+2;2ikr),$$
$$X=\Psi(\ell+1;2\ell+2;-2ikr)$$
(A.35)
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}-ik}{\beta_{\ell}+ik}}\right),$$
$$Y=\Psi(\ell+2;2\ell+3;-2ikr)$$
(A.36)
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}-ik}{\beta_{\ell}+ik}}\right),$$
$$M=\frac{\Gamma(\ell+1)k^{\ell+1}}{\left({\beta_{\ell}^{2}+k^{2}}\right)^{\ell+1}}$$
(A.37)
$${}\times\Psi(\ell+1;2\ell+2;-2ikr)-\frac{e^{-i\ell\pi}}{2^{2\ell+2}k^{\ell+1}}$$
$${}\times\Biggl{\{}\frac{2ik}{\left({\ell+1}\right)\left({\beta_{\ell}-ik}\right)}$$
$${}\times{}_{2}F_{1}\left(1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}\right)$$
$${}\times\Phi(\ell+1;2\ell+2;-2ikr)$$
$${}+\sum\limits_{n=0}^{\infty}\left({\frac{\beta_{\ell}+ik}{2ik}}\right)^{n}\frac{1}{\left({n!}\right)}$$
$${}\times\theta_{n+1}(\ell+1,2\ell+2;-2ikr)\Biggl{\}},$$
$$M_{1}=\left({-2ik}\right)\Biggl{[}-\frac{\Gamma(\ell+2)k^{\ell+1}}{\left({\beta_{\ell}^{2}+k^{2}}\right)^{\ell+1}}$$
(A.38)
$${}\times\Psi(\ell+2;2\ell+3;-2ikr)-\frac{e^{-i\ell\pi}}{2^{2\ell+2}k^{\ell+1}}$$
$${}\times\Biggl{\{}\frac{2ik}{\left({2\ell+2}\right)\left({\beta_{\ell}-ik}\right)}$$
$${}\times{}_{2}F_{1}\left(1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}\right)$$
$${}\times\Phi(\ell+2;2\ell+3;-2ikr)$$
$${}+\sum\limits_{n=0}^{\infty}\left({\frac{\beta_{\ell}+ik}{2ik}}\right)^{n}\frac{1}{\left({n!}\right)}$$
$${}\times n\theta_{n}(\ell+2,2\ell+3;-2ikr)\Biggl{\}}\Biggl{]},$$
$$P=\left[{\frac{\lambda_{\ell}\Gamma(2\ell+2)}{\left({\ell!}\right)^{2}\Gamma(\ell+2)}}\right]^{2}\frac{{}_{2}F_{1}\left({1,-\ell;\ell+2;\dfrac{\beta_{\ell}-ik}{\beta_{\ell}+ik}}\right){}_{2}F_{1}\left({1,-\ell;\ell+2;\dfrac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)}{2^{2\ell-2}\left({\beta_{\ell}^{2}+k^{2}}\right)\left[{D_{\ell}(k)}\right]^{2}}.$$
(A.39)
The Wronskian \(J_{\ell}(k,r)\) of the pair of the irregular
solutions \(f_{\ell\pm}(k,r)\) expressed in Eq. (A.32) in
conjunction with Eqs. (A.33)–(A.39) is well normalized as well as
conserved. Hence we get
$$\lim\limits_{r\to 0}J_{\ell}(k,r)=1$$
(A.40)
and
$$\lim\limits_{r\to\infty}J_{\ell}(k,r)=1.$$
(A.41)
The first derivative of the \(J_{\ell}(k,r)\) with respect to
\(r\) is obtained as
$$J^{\prime}_{\ell}(k,r)=k^{-1}\textrm{Im}\Big{[}\left({2k}\right)^{2\ell+2}r^{2\ell}$$
(A.42)
$${}\times\left\{{\left({2\ell+2}\right)ikr+\left({\ell+1}\right)\left({2\ell+1}\right)}\right\}A_{1}$$
$${}+i\left({2k}\right)^{2\ell+3}r^{2\ell+1}\left({ikr+\ell+1}\right)\left({\ell+1}\right)\left({A-A^{\ast}}\right)$$
$${}+i\left({2k}\right)^{2\ell+3}\left({\ell+1}\right)\left({2\ell+2}\right)r^{2\ell+1}A$$
$${}-\left({2k}\right)^{2\ell+4}\left({\ell+1}\right)r^{2\ell+2}\left\{{B\left({\ell+2}\right)-\left({\ell+1}\right)A_{2}}\right\}$$
$${}+\frac{\lambda_{\ell}\Gamma(2\ell+2)\left({2k}\right)^{\ell+1}r^{2\ell}}{\Gamma(\ell+2)\left({\ell!}\right)^{2}2^{\ell-1}}\left({T_{1}+T_{2}}\right)$$
$${}+\frac{\lambda_{\ell}\Gamma(2\ell+2)\left({2k}\right)^{\ell+1}r^{2\ell+1}}{\Gamma(\ell+2)\left({\ell!}\right)^{2}2^{\ell-1}}\left({T_{3}+T_{4}}\right)$$
$${}+P\left({\ell+1}\right)r^{2\ell}\big{\{}\left({2\ell+1}\right)MM^{\ast}$$
$${}+r\left({M^{\ast}M_{1}+MM_{1}^{\ast}}\right)\big{\}}+Pr^{2\ell+1}T_{5}\Big{]},$$
where
$$A_{2}=\Psi(\ell+2;2\ell+3;2ikr)$$
(A.43)
$${}\times\Psi(\ell+2;2\ell+3;-2ikr),$$
$$B=\Psi(\ell+1;2\ell+2;2ikr)$$
(A.44)
$${}\times\Psi(\ell+3;2\ell+4;-2ikr),$$
$$X_{1}=2ik\left({\ell+1}\right)\Psi(\ell+2;2\ell+3;-2ikr)$$
(A.45)
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}-ik}{\beta_{\ell}+ik}}\right),$$
$$Y_{1}=2ik\left({\ell+2}\right)\Psi(\ell+3;2\ell+4;-2ikr)$$
(A.46)
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}-ik}{\beta_{\ell}+ik}}\right),$$
$$M_{2}=\left({2ik}\right)^{2}\Biggl{[}\frac{\Gamma(\ell+3)k^{\ell+1}}{\left({\beta_{\ell}^{2}+k^{2}}\right)^{\ell+1}}$$
(A.47)
$${}\times\Psi(\ell+3;2\ell+4;-2ikr)-\frac{e^{{-i}\ell\pi}}{2^{2\ell+2}k^{\ell+1}}$$
$${}\times\Biggl{\{}\frac{2ik\left({\ell+2}\right)}{\left({2\ell+2}\right)\left({2\ell+3}\right)\left({\beta_{\ell}-ik}\right)}$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)$$
$${}\times\Phi(\ell+3;2\ell+4;-2ikr)$$
$${}+\sum\limits_{n=0}^{\infty}\left({\frac{\beta_{\ell}+ik}{2ik}}\right)^{n}\frac{1}{\left({n!}\right)}n\left({n-1}\right)$$
$${}\times\theta_{n-1}(\ell+3,2\ell+4;-2ikr)\Biggl{\}}\Biggl{]},$$
$$T_{1}=\{\left({2\ell+2}\right)ikr+\left({2\ell+1}\right)\left({\ell+1}\right)\}$$
(A.48)
$${}\times\left\{{\frac{MX^{\ast}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}+\frac{M^{\ast}X}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}}\right\},$$
$$T_{2}=r\left({ikr+\ell+1}\right)$$
(A.49)
$${}\times\left\{{\frac{MX_{1}^{\ast}+X^{\ast}M_{1}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}+\frac{M^{\ast}X_{1}+XM_{1}^{\ast}}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}}\right\},$$
$$T_{3}=\left({2\ell+2}\right)\biggl{\{}\frac{M_{1}X^{\ast}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}$$
(A.50)
$${}+\frac{2ik\left({\ell+1}\right)M^{\ast}Y}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}\biggl{\}},$$
$$T_{4}=r\biggl{\{}\frac{M_{2}X^{\ast}+X_{1}^{\ast}M_{1}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}$$
(A.51)
$${}+\frac{2ik\left({\ell+1}\right)\left({M^{\ast}Y_{1}+YM_{1}^{\ast}}\right)}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}\biggl{\}}$$
and
$$T_{5}=ik\left\{{\left({2\ell+2}\right)\mbox{ }MM^{\ast}+rMM_{1}^{\ast}}\right\}$$
(A.52)
$${}+\left({2\ell+2+ikr}\right)M^{\ast}M_{1}+r\left({M^{\ast}M_{2}+M_{1}M_{1}^{\ast}}\right).$$
The second derivative of the \(J_{\ell}(k,r)\) is
$$J^{\prime\prime}_{\ell}(k,r)=k^{-1}\textrm{Im}\Big{[}\left({2k}\right)^{2\ell+2}r^{2\ell-1}$$
(A.53)
$${}\times\left\{{\left({2\ell+1}\right)\left({2\ell+2}\right)ikr+2\ell\left({\ell+1}\right)\left({2\ell+1}\right)}\right\}A_{1}$$
$${}+2\left({2k}\right)^{2\ell+4}r^{2\ell+1}\left({\ell+1}\right)^{2}$$
$${}\times\left\{{\left({ikr+\ell+1}\right)+\left({2\ell+2}\right)}\right\}A_{2}$$
$${}+i\left({2k}\right)^{2\ell+3}r^{2\ell}\left[{T_{6}-T_{7}}\right]$$
$${}-\left({2k}\right)^{2\ell+4}r^{2\ell+1}\left({\ell+1}\right)\left({\ell+2}\right)T_{8}$$
$${}-i\left({\ell+1}\right)\left({2k}\right)^{2\ell+5}r^{2\ell+2}T_{9}$$
$${}+\frac{\lambda_{\ell}\Gamma(2\ell+2)\left({2k}\right)^{\ell+1}r^{2\ell-1}}{\Gamma(\ell+2)\left({\ell!}\right)^{2}2^{\ell-1}}\left[{T_{10}+T_{11}+T_{12}}\right]$$
$${}+\frac{\lambda_{\ell}\Gamma(2\ell+2)\left({2k}\right)^{\ell+1}r^{2\ell}}{\Gamma(\ell+2)\left({\ell!}\right)^{2}2^{\ell-1}}\left[{T_{13}+T_{14}+T_{15}}\right]$$
$${}+P\left({2\ell+1}\right)r^{2\ell-1}\!MM^{\ast}\!\left\{{2\ell\left({\ell+1}\right)+ikr\left({2\ell+2}\right)}\right\}$$
$${}+Pr^{2\ell}\left[{T_{16}+T_{17}}\right]+Pr^{2\ell+1}T_{18}$$
$${}+Pr^{2\ell+2}\left({M^{\ast}M_{3}+2M_{2}M_{1}^{\ast}+M_{1}M_{2}^{\ast}}\right)\Big{]}$$
with
$$C=\Psi(\ell+1;2\ell+2;2ikr)$$
(A.54)
$${}\times\Psi(\ell+4;2\ell+5;-2ikr),$$
$$D=\Psi(\ell+2;2\ell+3;2ikr)$$
(A.55)
$${}\times\Psi(\ell+3;2\ell+4;-2ikr),$$
$$X_{2}=\left({2ik}\right)^{2}\left({\ell+1}\right)\left({\ell+2}\right)$$
(A.56)
$${}\times\Psi(\ell+3;2\ell+4;-2ikr)$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}-ik}{\beta_{\ell}+ik}}\right),$$
$$Y_{2}=\left({2ik}\right)^{2}\left({\ell+2}\right)\left({\ell+3}\right)$$
(A.57)
$${}\times\Psi(\ell+4;2\ell+5;-2ikr)$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}-ik}{\beta_{\ell}+ik}}\right),$$
$$M_{3}=-\left({2ik}\right)^{3}\Biggl{[}-\frac{\Gamma(\ell+4)k^{\ell+1}}{\left({\beta_{\ell}^{2}+k^{2}}\right)^{\ell+1}}$$
(A.58)
$${}\times\Psi(\ell+4;2\ell+5;-2ikr)-\frac{e^{-i\ell\pi}}{2^{2\ell+2}k^{\ell+1}}$$
$${}\times\Biggl{\{}\frac{2ik\left({\ell+2}\right)\left({\ell+3}\right)}{\left({2\ell+2}\right)\left({2\ell+3}\right)\left({2\ell+4}\right)\left({\beta_{\ell}-ik}\right)}$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)$$
$${}\times\Phi(\ell+4;2\ell+5;-2ikr)$$
$${}+\sum\limits_{n=0}^{\infty}\left({\frac{\beta_{\ell}+ik}{2ik}}\right)^{n}\frac{1}{\left({n!}\right)}n\left({n-1}\right)\left({n-2}\right)$$
$${}\times\theta_{n-1}(\ell+4,2\ell+5;-2ikr)\Biggl{\}}\Biggl{]},$$
$$T_{6}=\Big{\{}\left({\ell+1}\right)\left({2\ell+1}\right)\left({2\ell+2}\right)$$
(A.59)
$${}+2\left\{{\left({\ell+1}\right)\left({2\ell+2}\right)ikr+\left({2\ell+1}\right)\left({\ell+1}\right)^{2}}\right\}\Big{\}}A,$$
$$T_{7}=2\Big{\{}\left({\ell+1}\right)\left({2\ell+2}\right)ikr$$
(A.60)
$${}+\left({2\ell+1}\right)\left({\ell+1}\right)^{2}\Big{\}}A^{\ast},$$
$$T_{8}=\left\{{\left({ikr+\ell+1}\right)+2\left({2\ell+2}\right)}\right\}B$$
(A.61)
$${}+\left({ikr+\ell+1}\right)B^{\ast},$$
$$T_{9}=\left({\ell+2}\right)\left({\ell+3}\right)C$$
(A.62)
$${}-2\left({\ell+1}\right)\left({\ell+2}\right)D+\left({\ell+1}\right)\left({\ell+2}\right)D^{\ast},$$
$$T_{10}=\big{\{}\left({2\ell+1}\right)\left({2\ell+2}\right)ikr$$
(A.63)
$${}+2\ell\left({\ell+1}\right)\left({2\ell+1}\right)\big{\}}$$
$${}\times\left\{{\frac{MX^{\ast}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}+\frac{M^{\ast}X}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}}\right\},$$
$$T_{11}=r\left\{{\left({4\ell+4}\right)ikr+\left({\ell+1}\right)\left({4\ell+2}\right)}\right\}$$
(A.64)
$${}\times\left\{{\frac{MX_{1}^{\ast}+X^{\ast}M_{1}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}+\frac{M^{\ast}X_{1}+XM_{1}^{\ast}}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}}\right\},$$
$$T_{12}=r^{2}\left({ikr+\ell+1}\right)$$
(A.65)
$${}\times\Biggl{\{}\frac{MX_{2}^{\ast}+2X_{1}^{\ast}M_{1}+X^{\ast}M_{2}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}$$
$${}+\frac{M^{\ast}X_{2}+2X_{1}M_{1}^{\ast}+XM_{2}^{\ast}}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}\Biggl{\}},$$
$$T_{13}=\left({2\ell+1}\right)\left({2\ell+2}\right)$$
(A.66)
$${}\times\left\{{\frac{M_{1}X^{\ast}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}+\frac{2ik\left({\ell+1}\right)M^{\ast}Y}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}}\right\},$$
$$T_{14}=r\left({4\ell+4}\right)\Biggl{\{}\frac{M_{2}X^{\ast}+X_{1}^{\ast}M_{1}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}$$
(A.67)
$${}+\frac{2ik\left({\ell+1}\right)\left({M^{\ast}Y_{1}+YM_{1}^{\ast}}\right)}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}\Biggl{\}},$$
$$T_{15}=r^{2}\Biggl{\{}\frac{M_{3}X^{\ast}+2X_{1}^{\ast}M_{2}+M_{1}X_{2}^{\ast}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}$$
(A.68)
$${}+\frac{2ik\left({\ell+1}\right)\left({M^{\ast}Y_{2}+2Y_{1}M_{1}^{\ast}+YM_{2}^{\ast}}\right)}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}\Biggl{\}},$$
$$T_{16}=\left\{{\left({\ell+1}\right)\left({4\ell+2}\right)+\left({2\ell+3}\right)ikr}\right\}$$
(A.69)
$${}\times\left({M^{\ast}M_{1}+MM_{1}^{\ast}}\right),$$
$$T_{17}=\left({2\ell+1}\right)\big{\{}ikrMM_{1}^{\ast}$$
(A.70)
$${}+\left({2\ell+2+ikr}\right)M^{\ast}M_{1}\big{\}}$$
and
$$T_{18}=\left({\ell+1}\right)\big{(}MM_{2}^{\ast}+2M_{1}M_{1}^{\ast}$$
(A.71)
$${}+M^{\ast}M_{2}\big{)}+\left({4\ell+4+ikr}\right)\left({M^{\ast}M_{2}+M_{1}M_{1}^{\ast}}\right)$$
$${}+ikr\left({MM_{2}^{\ast}+M_{1}M_{1}^{\ast}}\right).$$
From Eqs. (3) and (A.28) the quantity \(Q_{\ell}(k,r,s)\)reads as
$$Q_{\ell}(k,r,s)$$
(A.72)
$${}=k^{-1}\textrm{Im}\biggl{[}\left({2k}\right)^{2\ell+2}r^{\ell+1}s^{\ell+1}e^{-ikr}e^{iks}$$
$${}\times\Psi(\ell+1;2\ell+2;2ikr)\Psi(\ell+1;2\ell+2;-2iks)$$
$${}+\frac{\lambda_{\ell}\Gamma(2\ell+2)\left({2k}\right)^{\ell+1}}{\Gamma(\ell+2)\left({\ell!}\right)^{2}2^{\ell-1}}r^{\ell+1}s^{\ell+1}e^{-ikr}e^{iks}$$
$${}\times\biggl{\{}\frac{M(s)}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}\Psi(\ell+1;2\ell+2;2ikr)$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)$$
$${}+\frac{M^{\ast}}{D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}\Psi(\ell+1;2\ell+2;-2iks)$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}-ik}{\beta_{\ell}+ik}}\right)\biggl{\}}$$
$${}+Pr^{\ell+1}s^{\ell+1}e^{-ikr}e^{iks}M(s)M^{\ast}\biggl{]}$$
with
$$M(s)=\frac{\Gamma(\ell+1)k^{\ell+1}}{\left({\beta_{\ell}^{2}+k^{2}}\right)^{\ell+1}}$$
(A.73)
$${}\times\Psi(\ell+1;2\ell+2;-2iks)-\frac{e^{-i\ell\pi}}{2^{2\ell+2}k^{\ell+1}}$$
$${}\times\Biggl{\{}\frac{2ik}{\left({\ell+1}\right)\left({\beta_{\ell}-ik}\right)}{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)$$
$${}\times\Phi(\ell+1;2\ell+2;-2iks)$$
$${}+\sum\limits_{n=0}^{\infty}\left({\frac{\beta_{\ell}+ik}{2ik}}\right)^{n}\frac{1}{\left({n!}\right)}\theta_{n+1}(\ell+1,2\ell+2;-2iks)\Biggl{\}}.$$
Equation (A.72) in conjunction with Eqs. (A.29)–(A.31) leads to
$$Q^{\prime}_{\ell}(k,r,s)$$
(A.74)
$${}=k^{-1}\textrm{Im}\Big{[}\left({2k}\right)^{2\ell+2}r^{1}s^{\ell+1}e^{-ikr}e^{iks}$$
$${}\times\Psi(\ell+1;2\ell+2;2ikr)\Psi(\ell+1;2\ell+2;-2iks)$$
$${}\times\left({\ell+1-ikr}\right)-i\left({\ell+1}\right)\left({2k}\right)^{2\ell+3}$$
$${}\times r^{1+1}s^{\ell+1}e^{-ikr}e^{iks}\Psi(\ell+1;2\ell+2;-2iks)$$
$${}\times\Psi(\ell+2;2\ell+3;2ikr)+P_{2}s^{\ell+1}e^{iks}M(s)$$
$${}\times\Big{\{}\left({\ell+1-ikr}\right)\Psi(\ell+1;2\ell+2;2ikr)$$
$${}-2ikr\left({\ell+1}\right)\Psi(\ell+2;2\ell+3;2ikr)\Big{\}}$$
$${}+\frac{\lambda_{\ell}\Gamma(2\ell+2)\left({2k}\right)^{\ell+1}r^{\ell}s^{\ell+1}e^{-ikr}e^{iks}}{\Gamma(\ell+2)\left({\ell!}\right)^{2}2^{\ell-1}D_{\ell}^{\ast}(k)\left({\beta_{\ell}+ik}\right)}$$
$${}\times\Psi(\ell+1;2\ell+2;-2iks)$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}-ik}{\beta_{\ell}+ik}}\right)$$
$${}\times\left\{{\left({\ell+1-ikr}\right)M^{\ast}+rM_{1}^{\ast}}\right\}$$
$${}+Pr^{\ell}s^{\ell+1}e^{-ikr}e^{iks}M(s)$$
$${}\times\left\{{\left({\ell+1-ikr}\right)M^{\ast}+rM_{1}^{\ast}}\right\}\Big{]}$$
with
$$P_{2}=\frac{\lambda_{\ell}\Gamma(2\ell+2)\left({2k}\right)^{\ell+1}}{\Gamma(\ell+2)\left({\ell!}\right)^{2}2^{\ell-1}}$$
(A.75)
$${}\times\frac{r^{\ell}e^{-ikr}}{D_{\ell}(k)\left({\beta_{\ell}-ik}\right)}$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right).$$
Now the value of the definite integral \(\int_{0}^{\infty}{V(r,s)Q^{\prime}_{\ell}(k,r,s)}ds\) in Eq. (1) together with
Eqs. (A.74) and (A.75) and use of Eqs. (A.13), (A.14), (A.22), and
(A.25) leads to
$$\int\limits_{0}^{\infty}{V(r,s)Q^{\prime}_{\ell}(k,r,s)}ds$$
(A.76)
$${}=k^{-1}\textrm{Im }\Big{[}\lambda_{\ell}2^{-2\ell}\left({\ell!}\right)^{-2}r^{\ell}e^{-\beta_{\ell}r}$$
$${}\times\Big{[}P_{1}\big{\{}\left({\ell+1-ikr}\right)$$
$${}\times\Psi(\ell+1;2\ell+2;2ikr)$$
$${}-2ikr\left({\ell+1}\right)\Psi(\ell+2;2\ell+3;2ikr)\big{\}}$$
$${}+P_{2}\big{\{}\left({\ell+1-ikr}\right)\Psi(\ell+1;2\ell+2;2ikr)$$
$${}-2ikr\left({\ell+1}\right)\Psi(\ell+2;2\ell+3;2ikr)\big{\}}N$$
$${}+PNr^{\ell}e^{-ikr}\left\{{\left({\ell+1-ikr}\right)M^{\ast}+rM_{1}^{\ast}}\right\}$$
$${}+\lambda_{\ell}\left({\frac{\Gamma(2\ell+2)}{\Gamma(\ell+2)}}\right)^{2}P_{3}\big{\{}\left({\ell+1-ikr}\right)$$
$${}\times M^{\ast}+rM_{1}^{\ast}\big{\}}\Big{]}\Big{]},$$
where
$$P_{1}=\frac{\Gamma(2\ell+2)\left({2k}\right)^{2\ell+2}r^{\ell}e^{-ikr}}{\Gamma(\ell+2)\left({\beta_{\ell}-ik}\right)\left({-2ik}\right)^{2\ell+1}}$$
(A.77)
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)$$
and
$$P_{3}=\frac{\left({2k}\right)^{\ell+1}r^{\ell}e^{-ikr}}{\left({-2ik}\right)^{\ell+1}2^{\ell-1}\left({\ell!}\right)^{2}D_{\ell}(k)}$$
(A.78)
$${}\times\frac{1}{\left({\beta_{\ell}^{2}+k^{2}}\right)}$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}+ik}{\beta_{\ell}-ik}}\right)$$
$${}\times{}_{2}F_{1}\left({1,-\ell;\ell+2;\frac{\beta_{\ell}-ik}{\beta_{\ell}+ik}}\right).$$
Equations (1), (A.32), (A.42), (A.53), and (A.76) in conjunction
with Eqs. (A.33)–(A.39), (A.43)–(A.52), (A.54)–(A.71), and
(A.77), (A.78) produces the desired expression for equivalent
local potential \(V_{\ell}(k,r)\) for the Graz separable nonlocal
interaction.
