The Strong Scott Conjecture: the Density of Heavy Atoms Close to the Nucleus
in Book: Spectral Theory and Mathematical Physics 257-272 (2020).
We review what is known about the atomic density close to the nucleus of heavy atoms.
On the excess charge of a relativistic statistical model of molecules with an inhomogeneity correction
H. Chen, H. Siedentop
Journal of Physics A 53, 395201 (2020).
We show that the molecular relativistic Thomas–Fermi–Weizsäcker functional consisting of atoms of atomic numbers Z1, ..., Zk has a minimizer, if the particle number N is constrained to a number less or equal to the total nuclear charge Z := Z1 + ⋯ + ZK. 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.
Proof of the strong Scott conjecture for Chandrasekhar atoms
R.L. Frank, K. Merz, H. Siedentop, B. Simon
Pure and Applied Functional Analysis 5 (6), 1319 - 1356 (2020).
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.
Equivalence of Sobolev Norms Involving Generalized Hardy Operators
R.L. Frank, K. Merz, H. Siedentop
International Mathematics Research Notices 2021, 2284-2303 (2019).
We consider the fractional Schrödinger 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) Schrödinger operators and have an important application in the treatment of large relativistic atoms.
The Atomic Density on the Thomas-Fermi Length Scale for the Chandrasekhar Hamiltonian
K. Merz, H. Siedentop
Reports on Mathematical Physics 83, 387-391 (2019).
We consider a large neutral atom of atomic number Z, modelled by a pseudo-relativistic Hamiltonian of Chandrasekhar. We study its suitably rescaled 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 → ∞ when Z/c is fixed to a value not exceeding 2/π. This shows that the electron density on the Thomas–Fermi length scale does not exhibit any relativistic effects.