The Ground State Energy of Heavy Atoms According to Brown and Ravenhall: Absence of Relativistic Effects in Leading Order

It is shown that the ground state energy of heavy atoms is, to leading order, given by the non-relativistic Thomas-Fermi energy. The proof is based on the relativistic Hamiltonian of Brown and Ravenhall which is derived from quantum electrodynamics yielding energy levels correctly up to order $\alpha^2$Ry

Mueller's Exchange-Correlation Energy in Density-Matrix-Functional Theory

The increasing interest in the Mueller density-matrix-functional theory has led us to a systematic mathematical investigation of its properties. This functional is similar to the Hartree-Fock functional, but with a modified exchange term in which the square of the density matrix \gamma(X, X') is replaced by the square of \gamma^{1/2}(X, X'). After an extensive introductory discussion of density-matrix-functional theory we show, among other things, that this functional is convex (unlike the HF functional) and that energy minimizing \gamma's have unique densities \rho(x), which is a physically desirable property often absent in HF theory. We show that minimizers exist if N \leq Z, and derive various properties of the minimal energy and the corresponding minimizers. We also give a precise statement about the equation for the orbitals of \gamma, which is more complex than for HF theory. We state some open mathematical questions about the theory together with conjectured solutions.Comment: Latex, 42 pages, 1 figure. Minor error in the proof of Prop. 2 correcte

Scott correction for large atoms and molecules in a self-generated magnetic field

We consider a large neutral molecule with total nuclear charge $Z$ in non-relativistic quantum mechanics with a self-generated classical electromagnetic field. To ensure stability, we assume that Z\al^2\le \kappa_0 for a sufficiently small $\kappa_0$, where \al denotes the fine structure constant. We show that, in the simultaneous limit $Z\to\infty$, \al\to 0 such that \kappa =Z\al^2 is fixed, the ground state energy of the system is given by a two term expansion $c_1Z^{7/3} + c_2(\kappa) Z^2 + o(Z^2)$. The leading term is given by the non-magnetic Thomas-Fermi theory. Our result shows that the magnetic field affects only the second (so-called Scott) term in the expansion

Stability of atoms and molecules in an ultrarelativistic Thomas-Fermi-Weizsaecker model

We consider the zero mass limit of a relativistic Thomas-Fermi-Weizsaecker model of atoms and molecules. We find bounds for the critical nuclear charges that ensure stability.Comment: 8 pages, LaTe

Dipoles in Graphene Have Infinitely Many Bound States

We show that in graphene charge distributions with non-vanishing dipole moment have infinitely many bound states. The corresponding eigenvalues accumulate at the edges of the gap faster than any power

Stability of Relativistic Matter With Magnetic Fields

Stability of matter with Coulomb forces has been proved for non-relativistic dynamics, including arbitrarily large magnetic fields, and for relativistic dynamics without magnetic fields. In both cases stability requires that the fine structure constant alpha be not too large. It was unclear what would happen for both relativistic dynamics and magnetic fields, or even how to formulate the problem clearly. We show that the use of the Dirac operator allows both effects, provided the filled negative energy `sea' is defined properly. The use of the free Dirac operator to define the negative levels leads to catastrophe for any alpha, but the use of the Dirac operator with magnetic field leads to stability.Comment: This is an announcement of the work in cond-mat/9610195 (LaTeX