ASYMPTOTICS OF HEAVY-ATOMS IN HIGH MAGNETIC-FIELDS .2. SEMICLASSICAL REGIONS

The ground state energy of an atom of nuclear charge Ze in a magnetic field B is exactly evaluated to leading order as Z --> infinity in the following three regions: B much less than Z4/3, B approximately Z4/3 and Z4/3 much less than B much less than Z3. In each case this is acco mplished by a modified Thomas-Fermi (TF) type theory. We also analyze these TF theories in detail, one, of their consequences being the noni ntuitive fact that atoms are spherical (to leading order) despite the leading order change in energy due to the B field. This paper compleme nts and completes our earlier analysis [1], which was primarily devote d to the regions B approximately Z3 and B much greater than Z3 in whic h a semiclassical TF analysis is numerically and conceptually wrong. T here are two main mathematical results in this paper, needed for the p roof of the exactitude of the TF theories. One is a generalization of the Lieb-Thirring inequality for sums of eigenvalues to include magnet ic fields. The second is a semiclassical asymptotic formula for sums o f eigenvalues that is uniform in the field B.