Saddle-point model of a quantum point contact ($U=U_0-m^{\ast }{\omega }_x^2{x}^2/2+m^{\ast }{\omega }_y^2{y}^2/2$)

A locally separable potential with an unstable longitudinal maximum and stable transverse confinement. Mode $n$ sees barrier top $E_n=U_0+\hslash {\omega }_y(n+1/2)$ and transmission $T_n(E)=[1+\exp (-2\pi (E-E_n)/(\hslash {\omega }_x)){]}^{-1}$. It is a single-particle smooth-barrier model; interactions and bias-induced reshaping require additions.