453 research outputs found
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 Ry
The Energy of Heavy Atoms According to Brown and Ravenhall: The Scott Correction
We consider relativistic many-particle operators which - according to Brown
and Ravenhall - describe the electronic states of heavy atoms. Their ground
state energy is investigated in the limit of large nuclear charge and velocity
of light. We show that the leading quasi-classical behavior given by the
Thomas-Fermi theory is raised by a subleading correction, the Scott correction.
Our result is valid for the maximal range of coupling constants, including the
critical one. As a technical tool, a Sobolev-Gagliardo-Nirenberg-type
inequality is established for the critical atomic Brown-Ravenhall operator.
Moreover, we prove sharp upper and lower bound on the eigenvalues of the
hydrogenic Brown-Ravenhall operator up to and including the critical coupling
constant.Comment: 42 page
Equivalence of Sobolev norms involving generalized Hardy operators
We consider the fractional Schr\"odinger 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) Schr\"odinger operators and have an
important application in the treatment of large relativistic atoms.Comment: 16 pages; v2 contains improved results for positive coupling
constant
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
The Ground State Energy of Heavy Atoms: Relativistic Lowering of the Leading Energy Correction
We describe atoms by a pseudo-relativistic model that has its origin in the work of Chandrasekhar. We prove that the leading energy correction for heavy atoms, the Scott correction, exists. It turns out to be lower than in the non-relativistic description of atoms. 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 allows us to roll the proof back to results for the Schrödinger operator
Relativistic Strong Scott Conjecture: A Short Proof
We consider heavy neutral atoms of atomic number modeled with kinetic
energy used already by Chandrasekhar. We study the
behavior of the one-particle ground state density on the length scale
in the limit keeping 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
The Scott conjecture for large Coulomb systems: a review
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.Comment: 62 page
Scott correction for large atoms and molecules in a self-generated magnetic field
We consider a large neutral molecule with total nuclear charge 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 , where \al denotes the fine structure
constant. We show that, in the simultaneous limit , \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 . 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
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