Gisèle Ruiz Rieder

Thomas–Fermi theory with an external magnetic field

Of concern is a rigorous Thomas–Fermi theory of ground state electron densities for quantum mechanical systems in an external magnetic field. The energy functional takes the form E(ρ1,ρ2)=∑2i=1ci ∫R3ρi (x)5/3 dx + (1)/(2) ∫R3∫R3[ρ(x)ρ(y)/‖x−y‖]dx dy +∫R3V(x)ρ(x)dx +∫R3(B(x)(ρ1(x)−ρ2(x))dx; here ci is a positive constant, ρ1 [resp. ρ2] is the density of spin‐up [resp. spin‐down] electrons, ρ=ρ1+ρ2 is the total electron density, V is a given potential (typically a Coulomb potential describing electron–nuclear attraction), and B describes the effect of the external magnetic field. Let Ni=∫R3ρi(x)dx be the number of spin‐up and spin‐down electrons for i=1,2, and let N=N1+N2 be the total number of electrons. Specifying N and minimizing E(ρ1,ρ2) generally leads to a spin polarized system. For example, if B≤0 and B■0, then ρ1≥ρ2 and N1>N2. This and a number of related results are proved.

A nonlinear elliptic system arising in electron density theory

Of concern is the nonlinear system in R3 Here λi,Cj are given positive constants. Such a system arises in spin polarized ground state electron density theory. The parametres λ:1:, λ:2 should be such that where N1>,N2 are given positive numbers satisfying where Z is some give positive number. (Here N

A singular quasilinear parabolic problem in one space dimension

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The Fermi-Amaldi Correction in Spin Polarized Thomas-Fermi Theory

esub:, N

Spin‐polarized Thomas–Fermi theory

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A rigorous modified Thomas–Fermi theory for atomic systems

esub:, Z are, respectively, the number of spin up electrons, spin down electrons, and protons in the system.) We establish existence for this problem. Uniqueness remains open.

Continuous dependence of ergodic limits

Abstract Of concern are mixed initial boundary value problems of the form ∂u ∂t = ϑ (t, x, u, ∂u t6x ) ∂ 2 u t6x 2 , 0 ⩽ x ⩽ 1, 0 ⩽ t ⩽ T, where ϑ(t, x, p, q) ⩾ ϑ0(x) > 0 for 0

Recent rigorous results in Thomas-Fermi theory

Publisher Summary This chapter describes the Fermi–Amaldi correction in spin polarized Thomas–Fermi Theory. In Thomas–Fermi theory one seeks to find the ground state (electron position) density of a quantum mechanical N electron system by minimizing the energy functional. The spin polarized analogue of Thomas–Fermi theory. The chapter also derives the Euler–Lagrange equations. However, in this case no miracle occurs; the chapter obtains a coupled system of nonlinear elliptic equations which cannot be reduced to a single equation. The Euler–Lagrange equations are reduced to a coupled system of two elliptic partial differential equations.

The coulomb potential in higher dimensions

Of concern is a rigorous Thomas–Fermi theory of electron densities for spin‐polarized quantum‐mechanical systems. The number N↑, N↓ of spin‐up and spin‐down electrons are specified in advance, and one seeks to minimize the energy functional E(ρ↑,ρ↓) =c1∫R3(ρ↑(x)5/3 +ρ↓(x)5/3)dx +c2∫R3∫R3[ρ(x)ρ(y)/‖x −y‖]dx dy +∫R3V(x)ρ(x)dx, where c1, c2 are given positive constants, ρ↑ and ρ↓ are non‐negative functions, ρ=ρ↑ +ρ↓ is the total electron density, ∫R3ρ↑(x)dx =N↑, ∫R3ρ↓(x)dx =N↓, and V is a given potential. These results are analogous to the classical rigorous (spin‐unpolarized) Thomas–Fermi theory developed by Lieb and Simon [Phys. Rev. Lett. 33, 681 (1973)] and by Benilan and Brezis (‘‘The Thomas–Fermi problem,’’ in preparation).

Some extensions of Thomas-Fermi theory

Recently Parr and Ghosh [Proc. Natl. Acad. Sci. USA 83, 3577 (1986)] proposed a variant of the classical Thomas–Fermi theory of electrons in an atom. They produced a continuous electron density by introducing the constraint that the integral ∫R3 e−2k‖x‖ Δρ(x)dx exists, where k is determined by the nuclear cusp condition. Their results give improved calculations of ground state electron densities and energies. The present paper provides a rigorous mathematical foundation for the work of Parr and Ghosh and converts their results into theorems. Some generalizations are also obtained.