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Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions

1994; Springer Science+Business Media; Volume: 161; Issue: 1 Linguagem: Inglês

10.1007/bf02099414

1432-0916

Élliott H. Lieb, Jan Philip Solovej, Jakob Yngvason,

Quantum chaos and dynamical systems

The ground state energy of an atom of nuclear chargeZe in a magnetic fieldB is exactly evaluated to leading order asZ→∞ in the following three regions:B≪Z 4/3,B≈Z 4/3 andZ 4/3≪B≪Z3. In each case this is accomplished by a modified Thomas-Fermi (TF) type theory. We also analyze these TF theories in detail, one of their consequences being the nonintuitive fact that atoms are spherical (to leading order) despite the leading order change in energy due to theB field. This paper complements and completes our earlier analysis [1], which was primarily devoted to the regionsB≈Z 3 andB≫Z3 in which a semiclassical TF analysis is numerically and conceptually wrong. There are two main mathematical results in this paper, needed for the proof of the exactitude of the TF theories. One is a generalization of the Lieb-Thirring inequality for sums of eigenvalues to include magnetic fields. The second is a semiclassical asymptotic formula for sums of eigenvalues that isuniform in the fieldB.