Asymptotics of the Ground State Energy of Heavy Molecules in the Strong Magnetic Field

Abstract

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 my project deals mainly 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. Up to now 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 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 (this approach has provided very precise error estimate but works only in the very special cases).

Only recently M.Sigal and me applied semiclassical spectral asymptotics methods to the multiparticle quantum theory problems and justified the scott correction term for the ground state energy for large molecules. Automatically this provided some progress in other problems as well.

Now I am starting a big project which I call “Multiparticle Quantum Theory and Semiclassical Spectral Asymptotics”. The objective of this project is to solve the class of problems in semiclassical spectral asymptotics that arise in multiparticle quantum theory, using the machinery already developed or developing new machinery as appropriate. In the process this will integrate this theory into the toolbox of multiparticle quantum theory. Moreover, it will provide an extension of the best results obtained by separation of variables and WKB to general problems. For example, it provides a justification of the Dirac-schwinger correction for molecules, and it should lead to much improved estimates of the minimal distance between nuclei, and of the maximal charge of negative ions. My main emphasis will be on semiclassical spectral asymptotics theory backed by semiclassical microlocal analysis which are domains where I came from to MQT. The first step will be to consider heavy atoms and molecules in a strong magnetic field, which seems to be the most challenging problem of this type.