Heavy molecules in the strong magnetic field

We consider heavy atoms and molecules in the constant magnetic field under condition B <^ N where B is the intensity of the magnetic field and N is the number of electrons and discuss asymptotics of the ground state and ionization energies and estimates of the negative excessive charges for atoms and molecules and estimates of positive excessive charges for molecules. 0. Preface. Multiparticle quantum theory is one of the main topics of modern mathematical physics, and one of the central questions in this theory is the problem of the high-density limit. There are different versions of this problem including the analysis of a heavy atom, and the analysis of a molecule consisting of heavy atoms. These two versions are the most popular and I am dealing with them. The first step in the analysis is usually the Thomas-Fermi approximation, which leads to a non-linear partial differential system describing density and effective potential. This part of the theory is basically done. However, justification of this approximation, error estimates and the obtaining of additional correction terms (Scott and Dirac-Schwinger) is a much more difficult matter requiring quite different techniques. Until last years the main tool has been variational methods of mathematical physics. After no less than 20 years of intensive investigations there remain major open problems, and even recently essential progress was obtained. In some steps of the analysis there arise problems lying within the theory of semiclassical spectral asymptotics. This is a highly developed theory with the very strong machinery. However, problems specific for the multiparticle quantum theory have never been treated, and these problems have essential differences from standard problems of this theory. As a result these particular problems were treated either by variational methods as well (which led to non-accurate error estimate and the impossibility of recovering correction terms) or by separation of variables and investigation of ordinary differential equation by the WKB method + Work was partially supported by NSERC grant OGP0138277.