The nuclear energy density functionals with modified radial dependence of the isoscalar effective mass

Calculations for infinite nuclear matter with realistic nucleon-nucleon interactions suggest that the isoscalar effective mass (IEM) of a nucleon at the saturation density equals $m^{\ast }\mathrm{/}m\sim 0.8\pm 0.1$, at variance with empirical data on the nuclear level density in finite nuclei which are consistent with $m^{\ast }\mathrm{/}m\approx 1$. This contradicting results might be reconciled by enriching the radial dependence of IEM. In this work four new terms are introduced into the Skyrme-force inspired local energy-density functional: $\tau (\mathrm{\nabla }\rho {)}^2$, $\tau \frac{d\rho }{dr}$, ${\tau }^2$ and $\tau \mathrm{\Delta }\rho$. The aim is to investigate how they influence the radial dependence of IEM and, in turn, the single-particle spectra.