Energy eigenstates of spherically symmetric potentials using the shifted1Nexpansion

We show that an excellent analytic approximation to the energy eigenvalues and eigenfunctions of the Schrödinger equation can be obtained using the shifted $\frac{1}{N}$ expansion, where $N$ is the number of spatial dimensions. This technique, which was physically motivated for power-law potentials, is extended in this paper to general spherically symmetric potentials. The calculations are carried out for states with arbitrary quantum numbers $n$ and $l$ using fourth-order perturbation theory in the shifted expansion parameter $\frac{1}{\overline{k}}$, where $\overline{k}=N+2l-a$. We obtain very accurate agreement with numerical results for a variety of potentials for a very large range of both $n$ and $l$. Our results using the shift $a$ are consistently better than those previously obtained using the unshifted expansion parameter $\frac{1}{k}$, $k=N+2l$. The shifted $\frac{1}{N}$ expansion is seen to be applicable to a much wider class of problems than are most other approximation methods.