Finite square well

A finite square well is a bounded interval on which the potential is lower than outside, supporting matched oscillatory and exponential states.

For a zero-potential interval of width $a$, an infinite wall at its left end, and constant exterior potential $U_0$ to its right, the $j$th bound state branch exists precisely when $U_0>{\mathcal{U}}_j$, for $j=0,1,\dots$. Matching the wavefunction and its derivative gives $z\cot z=-\sqrt{z_0^2-z^2}$ with $z_0=a\sqrt{2m{U}_0}/\hslash$. At equality the new threshold solution is not normalizable, so exactly one bound state corresponds to $U_0\in ({\mathcal{U}}_0,{\mathcal{U}}_1]$.

For radii $0<a<b$, a hard-core spherical square-well potential has $V(r)=\infty$ for $r\le a$, $V(r)=-V_0$ for $a<r\le b$, and $V(r)=0$ for $r>b$. Its radial wavefunction vanishes at the hard-core radius and is matched continuously with its radial derivative at $r=b$.

For outer radius $b=2a$ and zero-energy well wavenumber ${\kappa }_0=\sqrt{2m{V}_0}/\hslash$, matching the S-wave scattering logarithmic derivative givesIt diverges when ${\kappa }_{0}a=(n+\frac{1}{2})\pi$, where the well has a zero-energy scattering resonance.