Simple eigenvalue formula for the Coulomb-plus-linear potential

The eigenvalues $E_{\mathrm{nl}}\left(\lambda \right)$ of the Hamiltonian $H=-\Delta -\frac{1}{r}+\lambda r$ are analyzed with the help of potential envelopes and kinetic potentials. The result is the following simple approximate eigenvalue formula: $\lambda =\frac{\{2\mathrm{}{\left(\nu E\right)}^3-\nu E^2[{(1+3\mathrm{}{\nu }^{2}E)}^{\frac{1}{2}}-1\left]\right\}}{\mu {[{(1+3\mathrm{}{\nu }^{2}E)}^{\frac{1}{2}}-1]}^3}$, where $E\ge \frac{-1\mathrm{}}{4{\nu }^2}$ is a lower bound to $E_{\mathrm{nl}}\left(\lambda \right)$ if $\nu =\mu =(n+l)\mathrm{}$, an upper bound if $\nu =\mu =(2n+l-\frac{1}{2})$, and a good approximation when $\nu =(n+l)\mathrm{}$ and $\mu =(1.794n+l-0.418)\mathrm{}$.