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

@article{AIHPC_2012__29_6_887_0,
     author = {Canc\`es, Eric and Stoltz, Gabriel},
     title = {A mathematical formulation of the random phase approximation for crystals},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {29},
     number = {6},
     year = {2012},
     pages = {887-925},
     doi = {10.1016/j.anihpc.2012.05.004},
     zbl = {1273.82073},
     mrnumber = {2995100},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2012__29_6_887_0}
}
Cancès, Eric; Stoltz, Gabriel. A mathematical formulation of the random phase approximation for crystals. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 6, pp. 887-925. doi : 10.1016/j.anihpc.2012.05.004. http://www.numdam.org/item/AIHPC_2012__29_6_887_0/

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