# A few words describing J.P. Solovej's research (last updated 2010)

My research deals with the rigorous mathematical study of physcial models. More specifically I have been interested in the stability and structure of atoms, molecules and macroscopic matter. Common to these systems is that they are described by quantum mechanical models, but the theory of relativity and questions from statistical mechanics enters also.

To study these problems I have specialized in the following areas of mathematics: analysis, functional analysis, ordinary and partial differential equations, spectral theory.

Here are examples of problems I have investigated in my research:

• Sizes of Atoms: Why are all atoms of comparable sizes?

For a discussion of the empirical radius of atoms see Mark Winther's Webelements

If you do not think that atoms are of realtively comparable sizes, consider the fact that Sodium (Na) with eleven electrons has essentally the same empirical radius as uranium (U) with 92 electrons (the empirical radius of Na is 180pm and the empirical radius of U is 175pm)

The papers [2,3,4,5,7,12,13,28,33] from my publication list are on this subject. One particular, point proved in [33] is that the radius of atoms may be well estimated by what is called Thomas-Fermi theory. The picture below illustrates this. The circles are empirical values of the radius whereas the solid curve is calculated using Thomas-Fermi theory

• Stability of Matter:

More details about this subject can be found in my review Stability of Matter, Encyclopedia of Mathematical Physics, eds. J.-P. Francoise, G.L. Naber and Tsou S.T. Oxford: Elsevier, 2006 (ISBN 978-0-1251-2666-3), volume 5 pages 8--14.

The papers [23,25,31,32,34,45,47,49,51,57,63,64] are on this subject.

• Magnetic fields: I have studied the structure of atoms, molecules and matter in strong magnetic fields. The papers [14,15,16,17,22,24,25,29,30,31,32,35,36,37,43,44] are on this subject.

• Semiclassical eigenvalue estimates: This mathematical subject is a useful tool in many applications in atomic, molecular, and condensed matter physics. The relevant papers are [9,17,20,27,29,30,41,42,43,44,65]

• Mean field approximations, Correlation estimates. Relevant papers: [18,19,21,23,26,33,49,55,58,60,61,62]

• Bose gases: Condensation and Bogolubov theory. This is explained in detail in the monograph Elliott H. Lieb, Robert Seiringer, J.P. Solovej, and J. Yngvason, The Mathematics of the Bose Gas and its Condensation. Series: Oberwolfach Seminars, Vol. 34, 2005, VIII, 208 p., ISBN: 3-7643-7336-9. A Birkhäuser textbook. Relevant papers are: [38,40,45,46,48,49,52,59]

For more details about my research consult all the papers I have written [publications] or look at copies of some of the transparencies that I have used when giving lectures at conferences [slides] (not updated since 2004).