Single particle level density in a finite depth potential well

We consider the single particle level density $g\left(\epsilon \right)$ of a realistic finite depth potential well, concentrating on the continuum $(\epsilon >0)$ region. We carry out quantum-mechanical calculations of the partial level density $g_l\left(\epsilon \right)$, associated with a well-defined orbital angular momentum $l<~40$, using the phase-shift derivative method and the Green’s-function method and compare the results with those obtained using the Thomas-Fermi approximation. We also numerically calculate $g\left(\epsilon \right)$ as a $l$ sum of $g_l\left(\epsilon \right)$ up to a certain value of ${\mathcal{l}}_{\max }<~40$ and determine the corresponding smooth level densities using the Strutinsky smoothing procedure. We demonstrate, in accordance with Levinson’s theorem, that the partial contribution $g_l\left(\epsilon \right)$ to the single particle level density from continuum states has positive and negative values. However, $g\left(\epsilon \right)$ is nonnegative. We also point out that this is not the case for an energy-dependent potential well.