IMA Postdoc Seminar (March 10, 2009)

Speaker: Heinz Siedentop (LMU Munich and IMA)

Title: The Ground State Energy of Heavy Atoms: Relativistic Lowering of the Leading Energy Correction

Abstract: Formulae for the ground state energy $E(Z)$ of atoms in the limit of large atomic number $Z$ have been a very active area of research. Among the most prominent contributions in the context of Schr\"odinger operators are the ground breaking work of Lieb and Simon (1977) and the seminal papers of Fefferman and Seco (1989-1994). However, large $Z$ results for non-relativistic models are questionable from a physical point of view: the innermost electrons of heavy atoms come close to the nucleus where they move very fast. Therefore, a relativistic model is important. We prove that the leading energy correction, the Scott correction -- intuitively due to the innermost electrons -- is indeed lowered compared with the Schr\"odinger case.

Our proof is valid up to and including the critical coupling constant. It is based on a renormalization of the energy whose zero level we adjust to be the ground-state energy of the corresponding non-relativistic problem. This -- together with a new Gagliaro-Nierenberg inequality -- allows us to roll the proof back to results for the Schr\"odinger operator.

The talk is based on joint work with Rupert Frank and Simone Warzel (both Princeton).