Symmetric Fermi-type potential

We utilize the amenability of the Fermi-type potential profile in Schr{"o}dinger equation to construct a symmetric one dimensional well as $V(x)=-U_n\mathrm{/}[1+\exp [(\mathrm{\mid }x\mathrm{\mid }-a)\mathrm{/}b]],U_n=V_n[1+\exp [-a\mathrm{/}b]]$. We define $\alpha =a\mathrm{/}b,{\beta }_n=b\sqrt{2m{U}_n}\mathrm{/}\mathrm{\hslash }$, we find ${\beta }_n$ values for which critically the well has $n$-node half bound state at $E=0$. Consequently, this fixed well has $n$ number of bound states. Also we obtain a semi-classical expression $\mathcal{G}(\alpha ,\beta )$ such that the Fermi well has either $[\mathcal{G}]$ or $[\mathcal{G}]+1$ number of bound states. Here $[\mathrm{.}]$ indicates the integer part. We also confirm the consistency of $\mathcal{G}$ with the number of s-wave neutron energy levels in a central ($x\in (0,\mathrm{\infty }))$ Fermi potential well.