Near-threshold quantization and level densities for potential wells with weak inverse-square tails

For potential tails consisting of an inverse-square term and an additional attractive ${1/r}^m$ term, $V\left(r\right)\mathrm{}\sim [{ħ}^2/(2\mathrm{}\mathcal{M}\left)\right]\left[\right(\gamma {/r}^2\left)-\right({\beta }^{m-2}{/r}^m\left)\right],$ we derive the near-threshold quantization rule $n=n\left(E\right)\mathrm{}$ which is related to the level density via $\rho =dn/dE.\mathrm{}$ For a weak inverse-square term, $-\frac{1}{4}<\gamma <\frac{3}{4}$ (and $m>\mathrm{}2),$ the leading contributions to $n\left(E\right)\mathrm{}$ are $n\stackrel{E\to 0}{=}A-B(-{E)}^{\sqrt{\gamma +1/4\mathrm{}}},$ so ρ has a singular contribution proportional to $(-{E)}^{\sqrt{\gamma +1/4\mathrm{}}-1}$ near threshold. The constant B in the near-threshold quantization rule also determines the strength of the leading contribution to the transmission probability through the potential tail at small positive energies. For $\gamma =0\mathrm{}$ we recover results derived previously for potential tails falling off faster than ${1/r}^2.$ The weak inverse-square tails bridge the gap between the more strongly repulsive tails, $\gamma >~3/4,$ where $n\left(E\right)\mathrm{}\stackrel{E\to 0}{=}A+O\left(E\right)\mathrm{}$ and ρ remains finite at threshold, and the strongly attractive tails, $\gamma <-1/4,$ where $n\stackrel{E\to 0}{=}B\ln (-E/A),$ which corresponds to an infinite dipole series of bound states and connects to the behavior $n\stackrel{E\to 0}{=}A+B{E}^{\mathrm{}(1/2)\mathrm{}-\mathrm{}(1/m)\mathrm{}},$ describing infinite Rydberg-like series in potentials with longer-ranged attractive tails falling off as ${1/r}^m,$ $0<m<\mathrm{}2.\mathrm{}$ For $\gamma =-1/4$ (and $m>\mathrm{}2)$ we obtain $n\left(E\right)\mathrm{}\stackrel{E\to 0}{=}A+C/\ln (-E/B),$ which remains finite at threshold.