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(rho-1,rho-2) = SIGMA-i2 = 1c(i) integral R3-rho-i(x)5/3 dx + 1/2 integral R3 integral R3[rho(x)rho(y)/\x - y\]dx dy + integral R3 V(x)rho(x)dx + integral R3 (B(x)(rho-1(x) - rho-2(x))dx; here c(i) is a positive constant, rho-1[resp. rho-2] is the density of spin-up [resp. spin-down] electrons, rho = rho-1 + rho-2 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 N(i) = integral R3 rho-i(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(rho-1,rho-2) generally leads to a spin polarized system. For example, if B less-than-or-equal-to 0 and B not-equal 0, then rho-1 greater-than-or-equal-to rho-2 and N1 > N2. This and a number of related results are proved.