Spatial densities of fermionic quantum systems typically exhibit oscillations that are connected to shell effects. We present a semiclassical theory [1] in which these oscillations are described in terms of the closed (but not periodic) orbits of the corresponding classical system. We also discuss the Friedel oscillations near a steep surface and present a case study of the two-dimensional circular billiard in which all periodic and non-periodic orbits can be classified through analytic relations [2]. Finally, we discuss local virial theorems that can be derived from this semiclassical theory [3].

[1] J. Roccia and M. Brack, Phys. Rev. Lett. 100, 200408 (2008);
J. Roccia, M. Brack, A. Koch, Phys. Rev. E 81, 011118 (2010).
[2] M. Brack and J. Roccia, J. Phys. A 42, 355210 (2009).
[3] M. Brack, A. Koch, M. V. N. Murthy, J. Roccia, J. Phys. A 43, 255204 (2010).