## 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).