Half-line finite well

A one-dimensional well $V=-V_0$ on $0<x<a$, with an infinite wall at zero and $V=0$ beyond $a$. For $-V_0<E<0$, the interior wavefunction is proportional to $\sin kx$ and the exterior decays as $e^{-qx}$. Continuity gives $k\cot ka=-q$, where $k^2=2m(E+V_0)/{\hslash }^2$ and $q^2=-2mE/{\hslash }^2$. Its half-line finite-well bound-state count has a nonzero minimum depth, unlike a full-line attractive one-dimensional well.

For dimensionless depth $\nu =a\sqrt{2m{V}_0}/\hslash$, each negative branch of $z\cot z=-\sqrt{{\nu }^2-z^2}$ gives one bound state. The $n$th state exists exactly when $\nu >(n-1/2)\pi$. At equality its exterior decay constant vanishes and the state is not square integrable. In particular, exactly two bound states means $3\pi /2<\nu \le 5\pi /2$. This strict lower and inclusive upper endpoint convention counts negative-energy eigenstates, excluding a zero-energy threshold solution.