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).
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