A parabolic quasi-Sturmian approach to quantum scattering by a Coulomb-like potential

Abstract

A computational method in parabolic coordinates is proposed to treat the scattering of a charged particle from both spherically and axially symmetric Coulomb-like potentials. Specifically, the long-range part of the Hamiltonian is represented in parabolic quasi-Sturmian basis functions, while the short-range part is approximated by a Sturmian $L^2$-basis-set truncated expansion. We establish an integral representation of the Coulomb Green’s function in parabolic coordinates from which we derive a convenient closed form for its matrix elements in the chosen $L^2$ basis set. From the Green’s function, we build quasi-Sturmian functions that are also given in closed form. Taking advantage of their adequate built-in Coulomb asymptotic behavior, scattering amplitudes are extracted as simple analytical sums that can be easily computed. The scheme, based on the proposed quasi-Sturmian approach, proves to be numerically efficient and robust as illustrated with converged results for three different scattering potentials, one of spherical and two of axial symmetry.

Appendices

The matrix elements of the Green’s function

In this appendix we provide the calculation details to perform the four-dimensional integral

$$\begin{aligned}&{\mathcal {G}}^{|m| (\pm )}_{n'_1 n'_2;\; n_1 n_2}(\beta , k)=\int \limits _{0}^{\infty }\hbox {d}\xi \int \limits _{0}^{\infty }\hbox {d}\eta \int \limits _{0}^{\infty }\hbox {d}\xi ' \int \limits _{0}^{\infty }\hbox {d}\eta ' \nonumber \\&\quad \varphi ^{|m|}_{n'_1}(\xi ) \varphi ^{|m|}_{n'_2}(\eta )\,{\mathcal {G}}^{|m|(\pm )}(k; \xi , \eta ; \xi ', \eta ') \varphi ^{|m|}_{n_1}(\xi ')\varphi ^{|m|}_{n_2}(\eta '). \end{aligned}$$

(67)

We first make use of the integral representation (17) and interchange integration orders. For the sake of brevity, we consider hereafter only the $(+)$ case. Consider first the integral over $\xi '$

$$\begin{aligned} {\mathcal {I}}_n(\xi )=\int \limits _{0}^{\infty }\hbox {d}\xi '\exp \left\{ i\frac{k}{2}\xi 'c-b\xi ' \right\} I_{\lambda }\left( -i k \sqrt{\xi \xi '}s\right) (2b\xi ')^{\frac{\lambda }{2}}L^{\lambda }_{n}(2b\xi '), \end{aligned}$$

(68)

where c and s denote $\cosh (z)$ and $\sinh (z)$, respectively. We calculate it using formula (2.19.12, 6) of Ref. [20]

$$\begin{aligned} {\mathcal {I}}_n(\xi )=\frac{(-1)^{n}}{b}\left( p s \sqrt{2x} \right) ^{\lambda } \frac{(1-pc)^{n}}{(1+pc)^{n+\lambda + 1}}\exp \left\{ \frac{p^2s^2x}{1+pc}\right\} L^{\lambda }_{n}\left( -2\frac{p^2s^2x}{1-p^2c^2} \right) , \end{aligned}$$

(69)

where $p= -i\frac{k}{2b}$, $x=b\xi$. Then we consider the integral over $\xi$ and perform the following auxiliary integral:

$$\begin{aligned} {\mathcal {J}}_{mn}=&\int \limits _{0}^{\infty }\hbox {d}\xi \left( 2 b \xi \right) ^{\frac{\lambda }{2}}L^{\lambda }_{m}(2 b \xi )\hbox {e}^{-b \xi }{\mathcal {I}}_n(\xi )\hbox {e}^{i\frac{k}{2}\xi c}\nonumber \\ =\,&(-1)^{n}\frac{2^{\lambda }}{b^2}(ps)^{\lambda }\frac{(1-pc)^{n}}{(1+pc)^{n+\lambda + 1}} \nonumber \\&\times \int \limits _{0}^{\infty }\hbox {d}x\, x^{\lambda }\exp \left\{ -x\frac{1+2pc+p^2}{1+pc} \right\} L^{\lambda }_{m}(2x)L^{\lambda }_{n}\left( -2\frac{p^2s^2x}{1-p^2c^2} \right) . \end{aligned}$$

(70)

Using formula (2.19.14, 6) of Ref. [20] we obtain

$$\begin{aligned} {\mathcal {J}}_{mn}=\,&(-1)^{m+n}\frac{2^{\lambda }}{b^2}\varGamma (\lambda +1)\,\frac{(\lambda +1)_{m}(\lambda +1)_{n}}{m! n!}\nonumber \\&\times \frac{(ps)^{\lambda } (1-p^2)^{m+n}}{(1+2pc+p^2)^{m+n+\lambda + 1}}\, {_2F_1}\left( -m, -n; \lambda +1;\frac{4p^2(c^2-1)}{(1-p^2)^2} \right) . \end{aligned}$$

(71)

The integration over $\eta$ and $\eta '$ are performed in exactly the same manner.

Further, for evaluating the Green’s function matrix element (67) it remains to integrate over z, and we perform the following integral:

$$\begin{aligned} {\mathcal {K}}=-i\mu k \int \limits _{0}^{\infty } \hbox {d}z\, \sinh (z)\left[ \coth \left( \frac{z}{2}\right) \right] ^{-2i\beta } {\mathcal {J}}_{n'_1,n_1}\,{\mathcal {J}}_{n'_2,n_2}. \end{aligned}$$

(72)

We use a change of variables

$$\begin{aligned} x = \left[ \tanh \left( \frac{z}{2}\right) \right] ^2 \Rightarrow \hbox {d}x=\sqrt{x}(1-x)\hbox {d}z \end{aligned}$$

(73)

to write

$$\begin{aligned} {\mathcal {K}}=\frac{i\mu }{2 k}\frac{1}{b^2}(1-\zeta ^2)^{2\lambda +2}(-\zeta )^{-K}(\lambda !)^2\frac{(\lambda +1)_{n'_1}(\lambda +1)_{n_1}}{n'_1! n_1!}\frac{(\lambda +1)_{n'_2}(\lambda +1)_{n_2}}{n'_2! n_2!}\ \mathcal {{\tilde{K}}}, \end{aligned}$$

(74)

where

$$\begin{aligned} \mathcal {{\tilde{K}}}=\,&\int \limits _{0}^{1} \hbox {d}x\, x^{i\beta + \lambda }(1-x)^K(1-\zeta ^2 x)^{-K-2\lambda -2} {_2F_1}\left( -n'_1, -n_1; \lambda +1; \left( \frac{1-\zeta ^2}{\zeta } \right) ^2\frac{x}{(1-x)^2} \right) \nonumber \\&\times {_2F_1}\left( -n'_2, -n_2; \lambda +1; \left( \frac{1-\zeta ^2}{\zeta } \right) ^2\frac{x}{(1-x)^2} \right) , \end{aligned}$$

(75)

and $K=n'_1+n'_2+n_1+n_2$. The integral $\mathcal {{\tilde{K}}}$ is calculated as follows. Using the expression of the polynomial

$$\begin{aligned} {_2F_1}\left( -m, -n; \lambda +1; z \right) =\sum \limits _{\ell = 0}^{\min (m, n)}\frac{\left( \begin{array}{c} m \\ {\ell } \end{array}\right) \left( \begin{array}{c} n\\ {\ell } \end{array}\right) }{\left( \begin{array}{c} {\lambda + \ell } \\ {\ell } \end{array}\right) }z^{\ell }, \end{aligned}$$

(76)

the product of the two hypergeometric functions becomes the polynomial

$$\begin{aligned} \sum \limits _{\ell =0}^{u+v} c_{\ell }\,\left[ \left( \frac{1-\zeta ^2}{\zeta } \right) ^2\frac{x}{(1-x)^2} \right] ^{\ell } \end{aligned}$$

(77)

whose coefficients $c_{\ell }$ are

$$\begin{aligned} c_{\ell }=\sum \limits _{j=\max (\ell -v,0)}^{\min (\ell , u)}\frac{ \left( \begin{array}{c} {n_1} \\ {j} \end{array}\right) \left( \begin{array}{c} {n'_1} \\ {j} \end{array}\right) \left( \begin{array}{c} {n_2} \\ {\ell -j} \end{array}\right) \left( \begin{array}{c} {n'_2} \\ {\ell -j} \end{array}\right) }{\left( \begin{array}{c} {j+\lambda } \\ {j} \end{array}\right) \left( \begin{array}{c} {\ell -j+\lambda } \\ {\ell -j} \end{array}\right) }, \end{aligned}$$

with $u=\min (n_1,n'_1 )$ and $v=\min (n_2, n'_2)$. We thus have

$$\begin{aligned} \mathcal {{\tilde{K}}}= \sum \limits _{\ell =0}^{u + v} c_{\ell }\,\left( \frac{1-\zeta ^2}{\zeta } \right) ^{2\ell } \,{\mathcal {L}}_{\ell } \end{aligned}$$

(78)

in terms of the integrals

$$\begin{aligned} {\mathcal {L}}_{\ell } = \int \limits _{0}^{1} \hbox {d}x\, x^{i\beta + \lambda + \ell }(1-x)^{K-2\ell }(1-\zeta ^2 x)^{-K-2\lambda -2} \end{aligned}$$

(79)

which are readily identified as the integral representation of Gauss hypergeometric functions [13] (Eq. (15.3.1))

$$\begin{aligned} {\mathcal {L}}_{\ell } =\,&\frac{\varGamma (i\beta + \lambda + \ell + 1)\varGamma (K-2\ell + 1)}{\varGamma (i\beta + K + \lambda + 2 -\ell )} \nonumber \\&\quad \times {_2F_1}\, \left( K + 2\lambda + 2, \lambda + \ell + 1 + i \beta ; K + \lambda + 2 -\ell + i \beta ; \zeta ^2 \right) \nonumber \\ =&\frac{1}{(1-\zeta ^2)^{2 \lambda + 1 + 2 \ell }} \frac{\varGamma (i\beta + \lambda + \ell + 1)\varGamma (K-2\ell + 1)}{\varGamma (i\beta + K + \lambda + 2 -\ell )} \nonumber \\&\quad \times {_2F_1}\left( K- 2 \ell + 1, -\lambda -\ell + i \beta ; K + \lambda + 2-\ell + i \beta ; \zeta ^2 \right) , \end{aligned}$$

(80)

the second equality being obtained through the transformation (15.3.3) of Ref. [13].

Inserting (80) and (78) into (74), and including the basis (19) normalization factors, we find that the matrix element (67) can be expressed as the finite sum (24).

Asymptotic behavior of the QS functions

In this appendix we explore the large-distance ($r=(\xi +\eta )/2 \rightarrow \infty$) behavior of the “radial” part ${\mathcal {P}}^{|m|(\pm )}_{n_1 n_2}(k;\xi , \eta )$ of the QS functions, starting from the integral representation (30). For the sake of brevity we present here the $(+)$ results only; the $(-)$ case is obtained by complex conjugation. Let us express the Laguerre polynomials $L_n^{\lambda }$ in (30) as

$$\begin{aligned} L_{n}^{\lambda }(x)= \sum \limits _{\nu =0}^{n}c_{\nu }^{(n, \lambda )}x^{\nu }, \end{aligned}$$

(81)

where the coefficients $c_{\nu }^{(n, \lambda )}$ are given by (33) [13]. Thus, the integral in (30) is expressed in terms of the integrals

$$\begin{aligned} {\mathcal {I}}_{{\mathcal {M}}, {\mathcal {N}}} = \int \limits _{0}^{1} \hbox {d}s (1 - s)^{i \beta + {\mathcal {M}}}(1 - \zeta s)^{-i \beta + {\mathcal {M}}}(1 - s - \zeta s)^{{\mathcal {N}}}\hbox {e}^{\alpha s}, \end{aligned}$$

(82)

where

$$\begin{aligned} {\mathcal {M}} = \lambda + \nu _1 + \nu _2, \qquad {\mathcal {N}} = n_1 + n_2 - \nu _1 - \nu _2, \end{aligned}$$

(83)

and

$$\begin{aligned} \alpha = 2b(\xi + \eta )\frac{\zeta }{\zeta + 1}. \end{aligned}$$

(84)

Then, using the formal binomial expansions

$$\begin{aligned} \left( 1-\zeta s\right) ^{-i \beta +{\mathcal {M}}}=\sum \limits _{\ell = 0}^{\infty }\frac{\varGamma (-i \beta + {\mathcal {M}} + 1)}{\varGamma (-i \beta + {\mathcal {M}} + 1 - \ell )}\frac{(-\zeta s)^{\ell }}{\varGamma (\ell + 1)} \end{aligned}$$

(85)

and

$$\begin{aligned} (1 - s - \zeta s)^{{\mathcal {N}}} = \sum \limits _{\nu = 0}^{{\mathcal {N}}} \left( \begin{array}{c} {{\mathcal {N}}}\\ {{\mathcal {N}}-\nu } \end{array}\right) (1 - s)^{\nu }(-\zeta s)^{{\mathcal {N}}-\nu }, \end{aligned}$$

(86)

the integral (82) is given by a sum of integrals

$$\begin{aligned} {\mathcal {J}}_{{\mathcal {M}} + \nu , {\mathcal {N}}+\ell -\nu }&\equiv \int \limits _{0}^{1} \hbox {d}s (1 - s)^{i \beta +{\mathcal {M}} + \nu }\,s^{{\mathcal {N}} + \ell -\nu }\,\hbox {e}^{\alpha s}\nonumber \\&=\frac{\varGamma (i \beta +{\mathcal {M}} + \nu + 1)\varGamma ({\mathcal {N}} + \ell -\nu + 1)}{\varGamma ( i \beta + {\mathcal {M}} + {\mathcal {N}} + \ell + 2 )}\nonumber \\&\quad \times M({\mathcal {N}} + \ell -\nu + 1, i \beta + {\mathcal {M}} + {\mathcal {N}} + \ell + 2; \alpha ) \end{aligned}$$

(87)

that are easily identified to be related to the Kummer function M (second equality). Using the asymptotic approximation of the Kummer function for large $|\alpha |$ (see, e.g., Eq. (13.5.1) in Ref. [13]), we find the following $(\ell$- and ${\mathcal {N}}$-independent) asymptotic expression:

$$\begin{aligned} {\mathcal {J}}_{{\mathcal {M}} + \nu , {\mathcal {N}} + \ell -\nu } \simeq \varGamma (i \beta + {\mathcal {M}} + \nu + 1)\, \alpha ^{-(i \beta + {\mathcal {M}} + \nu + 1)}\,\hbox {e}^{\alpha }. \end{aligned}$$

(88)

As a consequence, one can retain (for fixed ${\mathcal {M}}$) only the $\nu =0$ term in (86), and therefore the integral (82) is approximated by

$$\begin{aligned} {\mathcal {I}}_{{\mathcal {M}}, {\mathcal {N}}} \simeq (-\zeta )^{{\mathcal {N}}}(1 - \zeta )^{-i \beta + {\mathcal {M}} }\, \varGamma (i \beta + {\mathcal {M}} + 1)\, \alpha ^{-(i \beta + {\mathcal {M}} + 1)}\,\hbox {e}^{\alpha }. \end{aligned}$$

(89)

Finally, after some algebra and using the fact that $\sqrt{\xi \eta }=\sqrt{x^2+y^2}=r \sin \theta =(1/2) (\xi +\eta ) \sin \theta$, we obtain

$$\begin{aligned} {\mathcal {P}}^{\lambda (+)}_{n_1 n_2}(k; \xi , \eta ) \underset{r\rightarrow \infty }{\longrightarrow } {\mathcal {A}}_{n_1 n_2}^{\lambda (+)}(\theta )\, \frac{\exp \left\{ i \left[ kr-\beta \ln (2kr)\right] \right\} }{r}, \end{aligned}$$

(90)

with the amplitudes ${\mathcal {A}}^{\lambda (+)}_{n_1 n_2}(\theta )$ given by Eq. (32). For the $(-)$ case we simply have ${\mathcal {A}}_{n_1 n_2}^{\lambda (-)}(\theta )=\left[ {\mathcal {A}}_{n_1 n_2}^{\lambda (+)}(\theta ) \right] ^*$.