Local virial theorems and closed-orbit theory for spatial density oscillations in fermionic systems

We investigate the particle and kinetic energy densities for a system of $N$ fermions confined in a local mean-field potential $V(\bfr)$. For isotropic harmonic oscillators in arbitrary dimensions, exact linear relations between kinetic and potential energy density, termed ``local virial theorems'', and some exact (integro-) differential equations for the particle density have been earlier derived. Here we show the same relations to hold for linear potentials in arbitrary dimensions, and some of them for the one-dimensional box. We then use a recently developed semiclassical theory for density oscillations [J. Roccia and M. Brack, Phys.\ Rev.\ Lett.\ {\bf 100}, 200408 (2008)] to generalize these theorems to arbitrary potentials and test their validity for various non-harmonic potentials. We also discuss the relevance of our results for density functional theory. We show, in particular, that the Thomas-Fermi functional $\tau_{\text{TF}}[\rho]$ for the suitably defined kinetic energy density $\tau(\bfr)$ reproduces the quantum shell oscillations correctly to leading order in the oscillating parts