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This question was already asked here and this is a suitable form of the equation for numerically solving. \begin{align*} i\frac{\partial}{\partial t}u_{\ell}(r,t) = \Bigg(-\frac{1}{2} \frac{\partial ^2}{\partial r^2} + \frac{\ell(\ell+1)}{2r^2} + V(r)\Bigg)u_{\ell}(r,t)\quad \quad \quad \quad \quad \quad \quad \nonumber\\ + \frac{2}{3}k(t)r^2 \sum_{\ell' = {max}(\ell-2,0)}^{{min}(\ell+2,L_{{max}})} \Bigg(\delta_{\ell,\ell'} - \sqrt{ \frac{4\pi}{5}} \alpha(\ell, \ell') \Bigg) u_{\ell'}(r,t) \end{align*} where the coefficient $\alpha(\ell, \ell')$ is given by \begin{align*} \alpha(\ell, \ell') = \int Y^*_{\ell0}(\Omega)Y_{20}(\Omega)Y_{\ell'0}(\Omega) d\Omega \end{align*}

which the integral can be solved by Wigner-3j coefficients (see here ).

Can anyone suggest an efficient numerically method for solving the equation?

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  • $\begingroup$ Should the last term of the equation involve $u_{\ell'}$ instead of $u_\ell$? As stated, the equations for different values of $\ell$ do not couple. $\endgroup$ Jul 7, 2017 at 23:20

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