Many observables or functions describing a system of $N$ interacting fermions can be
separated into smooth and oscillating components. I will briefly discuss two different
types of semiclassical (or asymptotic) expansions that can be used for the two components.
I review the extended Thomas-Fermi (ETF) expansion of the smooth components and discuss
the corresponding functionals $T_s[\rho]$ and $\tau[\rho]$ for the smooth kinetic energy.
Then I briefly review the periodic orbit theory (POT) for level density and energy, that
relates the quantum oscillations to the periodic orbits of the corresponding classical
system by means of so-called "trace formulae".
In the main part of my talk, I present a recently developed semiclassical theory [1] for
the oscillations in spatial densities $\rho(r)$, $\tau(r)$, etc. Based upon its results,
we can show that the Thomas-Fermi functional $\tau_{TF}[\rho]$, when used with the exact
quantum-mechanical density $\rho(r)$, is able to reproduce the quantum oscillations in
the kinetic-energy density $\tau(r)$ to a high degree of accuracy. This is due to a
"local virial theorem" (LVT) [1,2] that relates kinetic and potential energy densities
locally at any point in space and is asymptotically exact in the semiclassical limit
$\hbar \to 0$. For linear and harmonic potentials, a generalized LVT has been shown to
hold asymptotically for the limit $N \to \infty$ amongst the exact quantum-mechanical
densities $\rho(r)$ and $\tau(r)$.

[1] J. Roccia, M. Brack and A. Koch, Phys. Rev. E 81, 011118 (2010), and earlier
references quoted therein.

[2] M. Brack, A. Koch, M. V. N. Murthy and J. Roccia, J. Phys. A 43, 255204 (2010).