no. 6
A mathematical formulation of the random phase approximation for crystals
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, pp. 887-925

This works extends the recent study on the dielectric permittivity of crystals within the Hartree model [E. Cancès, M. Lewin, Arch. Ration. Mech. Anal. 197 (1) (2010) 139–177] to the time-dependent setting. In particular, we prove the existence and uniqueness of the nonlinear Hartree dynamics (also called the random phase approximation in the physics literature), in a suitable functional space allowing to describe a local defect embedded in a perfect crystal. We also give a rigorous mathematical definition of the microscopic frequency-dependent polarization matrix, and derive the macroscopic Maxwell–Gauss equation for insulating and semiconducting crystals, from a first order approximation of the nonlinear Hartree model, by means of homogenization arguments.

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     title = {A mathematical formulation of the random phase approximation for crystals},
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Cancès, Eric; Stoltz, Gabriel. A mathematical formulation of the random phase approximation for crystals. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, pp. 887-925. doi: 10.1016/j.anihpc.2012.05.004

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