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:
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
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.
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).