Counting the Bound States for Central Potentials

An interaction potential is fitted by a multiple-step function, step width $\Delta R\to 0$, to yield a first-order differential equation for phase function ${\eta }_l\mathrm{}(k,r)\mathrm{}$, and also an integral for the wave-function amplitude $A_l\mathrm{}(k,r)\mathrm{}$. Both neutral- and charged-particle scattering is treated. The number of bound states follows by integration of the ${\eta }_l\mathrm{}(0,r)\mathrm{}$ equation and Levinson's theorem ${\eta }_l(0,\infty )=n_l\pi$; also, for $U^{\prime }(r\ge R)\ge 0$, one has $\left|{\psi }_0\right(r\ge R\left)\right|\le 1$. An approximate method of evaluating all the eigenenergies via scattering phase shifts is discussed.