We review some older and more recent results concerning the energy and particle distribution in ground states of heavy Coulomb systems. The reviewed results are asymptotic in nature: they describe properties of many-particle systems in the limit of a large number of particles. Particular emphasis is put on models that take relativistic kinematics into account. While non-relativistic models are typically rather well understood, this is generally not the case for relativistic ones and leads to a variety of open questions.

We consider the fractional Schrodinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces, which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schrodinger operators and have an important application in the treatment of large relativistic atoms.

We give the asymptotic behavior of the ground state energy of Engel's and Dreizler's relativistic Thomas-Fermi-Weizsäcker-Dirac functional for heavy atoms for fixed ratio of the atomic number and the velocity of light. Using a variation of the lower bound, we show stability of matter.

We review what is known about the atomic density close to the nucleus of heavy atoms.

We show that the molecular relativistic Thomas-Fermi-Weizsacker functional consisting of atoms of atomic numbersZ(1), horizontal ellipsis ,Z(k)has a minimizer, if the particle numberNis constrained to a number less or equal to the total nuclear chargeZ colon equals Z(1)+ MIDLINE HORIZONTAL ELLIPSIS +Z(K). Moreover, there is no minimizer, if the particle number exceeds 2.56Z. This gives lower and upper bounds on the maximal ionization of heavy atoms.

We consider heavy neutral atoms of atomic number Z modeled with kinetic energy (c^2p^2+c^4)^1/2−c^2 used already by Chandrasekhar. We study the behavior of the one-particle ground state density on the length scale Z−1 in the limit Z,c→∞ keeping Z/c fixed. We give a short proof of a recent result by the authors and Barry Simon showing the convergence of the density to the relativistic hydrogenic density on this scale.

We prove the convergence of the density on the scale Z−1 to the density of the Bohr atom (with infinitely many electrons) (strong Scott conjecture) for a model that is known to describe heavy atoms accurately.

We consider a large neutral atom of atomic number Z, taking relativistic effects into account by assuming the dispersion relation √(c^2p^2+c^4). We study the behavior of the one-particle ground state density on the length scale Z−1 in the limit Z,c→∞ keeping Z/c fixed and find that the spherically averaged density as well as all individual angular momentum densities separately converge to the relativistic hydrogenic ones. This proves the generalization of the strong Scott conjecture for relativistic atoms and shows, in particular, that relativistic effects occur close to the nucleus. Along the way we prove upper bounds on the relativistic hydrogenic density.

We consider a large neutral atom of atomic number Z, modelled by a pseudo-relativistic Hamiltonian of Chandrasekhar. We study its suitably resealed one-particle ground state density on the Thomas-Fermi length scale Z(-1/3). Using an observation by Fefferman and Seco, we find that the density on this scale converges to the minimizer of the Thomas-Fermi functional of hydrogen as Z -> infinity when Z/c is fixed to a value not exceeding 2/pi. This shows that the electron density on the Thomas-Fermi length scale does not exhibit any relativistic effects.