Eigenvalue Method

The eigenvalues, $E_{\mathrm{nl}}$, of the radial Schrödinger equation are approximated by $c_k(E=E_{\mathrm{nl}})=0\mathrm{}$ ($k=2, 3, \dots \infty$), where $E$ is the energy and the $c_k$ are the power-series coefficients of $f\left(\mathcal{r}\right)=\frac{\psi \left(\mathcal{r}\right)}{\left[{\mathcal{r}}^{l+1\mathrm{}}w\right(\mathcal{r}\left)\right]}$ with $\psi \left(\mathcal{r}\right)$ the wave function and $w\left(\mathcal{r}\right)$ a weight function. This is an extension of the method used to solve certain potentials exactly where $c_k(E=E_{\mathrm{nl}})=0\mathrm{}$ terminates a power series.