On Effective PDEs of Quantum Physics

On the Reduced Hartree-Fock Equations with a Small Anderson Type Background Charge Distribution

“…In condensed matter physics, the density of electrons is governed by the Kohn-Sham (KS) equation of DFT, whose mathematical properties have become an area of intense research, for example, [1,3,4,5,6,7,8,9,11,10,14,15,16,17,18,19,20]. One central focus of these studies is the existence and uniqueness problem.…”

“…A notable example is when κ is the Anderson type disorder. At zero temperature, some mitigation of the long range Coulomb problem involves studying minimizers of E KS on a finite domain and then passing to their thermal dynamical limits to include periodic κ and ρ γ (see for example, [6,7,9]), or introducing a screening mass as in the case of Yukawa potential [4]. On the other hand, a natural screening is observed at positive temperature [8,14].…”

On a Novel Effective Equation of the Reduced Hartree-Fock Theory

“…Since this type of existence result is well studied in the literature, we will not enumerate all previous works. The interested reader is referred to, for example, [1,9,10,12,32]. (35) for definition of κ 0 ), there exists a solution (φ, µ) ∈ H 2 (Ω; R) × R to the LSC equation (30).…”