Symmetric Fermi-type potential

Abstract

We utilize the amenability of the Fermi-type potential profile in Schr{\"o}dinger equation to construct a symmetric one dimensional well as $V(x){=}{-}U_n/[1+\exp[(|x|{-}a)/b]], ~ U_n{=}V_n[1+\exp[-a/b]]$. We define $\alpha=a/b, ~\beta_n {=}b\sqrt{2m U_n}/\hbar$, 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 ${\cal G}(\alpha,\beta)$ such that the Fermi well has either $[\cal G]$ or $[{\cal G}]+1$ number of bound states. Here $[.]$ indicates the integer part. We also confirm the consistency of $\cal G$ with the number of s-wave neutron energy levels in a central ($x\in (0,\infty))$ Fermi potential well.