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Spectral characteristics for a spherically confined $-a/r+b{r}^{2}$ potential. (English) Zbl 1215.81144
Summary: We consider the analytical properties of the eigenspectrum generated by a class of central potentials given by $V\left(r\right)=-a/r+b{r}^{2}$, $b>0$. In particular, scaling, monotonicity, and energy bounds are discussed. The potential $V\left(r\right)$ is considered both in all space, and under the condition of spherical confinement inside an impenetrable spherical boundary of radius $R$. With the help of the asymptotic iteration method, several exact analytic results are obtained which exhibit the parametric dependence of energy on $a$, $b$, and $R$, under certain constraints. More general spectral characteristics are identified by use of a combination of analytical properties and accurate numerical calculations of the energies, obtained by both the generalized pseudo-spectral method, and the asymptotic iteration method. The experimental significance of the results for both the free and confined potential $V\left(r\right)$ cases are discussed.
##### MSC:
 81V45 Applications of quantum theory to atomic physics 81Q05 Closed and approximate solutions to quantum-mechanical equations 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 39B12 Iterative and composite functional equations 81T80 Simulation and numerical modelling (quantum field theory)