Gradient corrections for semiclassical theories of atoms in strong magnetic fields

This paper is divided into two parts. In the first one the von Weizs\"acker term is introduced to the Magnetic TF theory and the resulting MTFW functional is mathematically analyzed. In particular, it is shown that the von Weizs\"acker term produces the Scott correction up to magnetic fields of order $B \ll Z^2$, in accordance with a result of V. Ivrii on the quantum mechanical ground state energy. The second part is dedicated to gradient corrections for semiclassical theories of atoms restricted to electrons in the lowest Landau band. We consider modifications of the Thomas-Fermi theory for strong magnetic fields (STF), i.e. for $B \ll Z^3$. The main modification consists in replacing the integration over the variables perpendicular to the field by an expansion in angular momentum eigenfunctions in the lowest Landau band. This leads to a functional (DSTF) depending on a sequence of one-dimensional densities. For a one-dimensional Fermi gas the analogue of a Weizs\"acker correction has a negative sign and we discuss the corresponding modification of the DSTF functional.Comment: Latex2e, 36 page

An equivalence relation of boundary/initial conditions, and the infinite limit properties

The 'n-equivalences' of boundary conditions of lattice models are introduced and it is derived that the models with n-equivalent boundary conditions result in the identical free energy. It is shown that the free energy of the six-vertex model is classified through the density of left/down arrows on the boundary. The free energy becomes identical to that obtained by Lieb and Sutherland with the periodic boundary condition, if the density of the arrows is equal to 1/2. The relation to the structure of the transfer matrix and a relation to stochastic processes are noted.Comment: 6 pages with a figure, no change but the omitted figure is adde

Quantum shock waves in the Heisenberg XY model

We show the existence of quantum states of the Heisenberg XY chain which closely follow the motion of the corresponding semi-classical ones, and whose evolution resemble the propagation of a shock wave in a fluid. These states are exact solutions of the Schroedinger equation of the XY model and their classical counterpart are simply domain walls or soliton-like solutions.Comment: 15 pages,6 figure

A nonlinear indentity for the scattering phase of integrable models

A nonlinear identity for the scattering phase of quantum integrable models is proved.Comment: 5 pages, Latex, no figure

The Ground States of Large Quantum Dots in Magnetic Fields

The quantum mechanical ground state of a 2D $N$-electron system in a confining potential $V(x)=Kv(x)$ ($K$ is a coupling constant) and a homogeneous magnetic field $B$ is studied in the high density limit $N\to\infty$, $K\to \infty$ with $K/N$ fixed. It is proved that the ground state energy and electronic density can be computed {\it exactly} in this limit by minimizing simple functionals of the density. There are three such functionals depending on the way $B/N$ varies as $N\to\infty$: A 2D Thomas-Fermi (TF) theory applies in the case $B/N\to 0$; if $B/N\to{\rm const.}\neq 0$ the correct limit theory is a modified $B$-dependent TF model, and the case $B/N\to\infty$ is described by a ``classical'' continuum electrostatic theory. For homogeneous potentials this last model describes also the weak coupling limit $K/N\to 0$ for arbitrary $B$. Important steps in the proof are the derivation of a new Lieb-Thirring inequality for the sum of eigenvalues of single particle Hamiltonians in 2D with magnetic fields, and an estimation of the exchange-correlation energy. For this last estimate we study a model of classical point charges with electrostatic interactions that provides a lower bound for the true quantum mechanical energy.Comment: 57 pages, Plain tex, 5 figures in separate uufil

Spin of the ground state and the flux phase problem on the ring

As a continuation of our previous work, we derive the optimal flux phase which minimizes the ground state energy in the one-dimensional many particle systems, when the number of particles is odd in the absence of on-site interaction and external potential. Moreover, we study the relationship between the flux on the ring and the spin of the ground state through which we derive some information on the sum of the lowest eigenvalues of one-particle Hamiltonians

Exponential localization of hydrogen-like atoms in relativistic quantum electrodynamics

We consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. We prove that the no-pair operator is semi-bounded below and that its spectral subspaces corresponding to energies below the ionization threshold are exponentially localized. Both results hold true, for arbitrary values of the fine-structure constant, $e^2$, and the ultra-violet cut-off, $\Lambda$, and for all nuclear charges less than the critical charge without radiation field, $Z_c=e^{-2}2/(2/\pi+\pi/2)$. We obtain similar results for the semi-relativistic Pauli-Fierz operator, again for all values of $e^2$ and $\Lambda$ and for nuclear charges less than $e^{-2}2/\pi$.Comment: 37 page

Improved Lieb-Oxford exchange-correlation inequality with gradient correction

We prove a Lieb-Oxford-type inequality on the indirect part of the Coulomb energy of a general many-particle quantum state, with a lower constant than the original statement but involving an additional gradient correction. The result is similar to a recent inequality of Benguria, Bley and Loss, except that the correction term is purely local, which is more usual in density functional theory. In an appendix, we discuss the connection between the indirect energy and the classical Jellium energy for constant densities. We show that they differ by an explicit shift due to the long range of the Coulomb potential.Comment: Final version to appear in Physical Review A. Compared to the very first version, this one contains an appendix discussing the link with the Jellium proble

On the Lieb-Thirring constants L_gamma,1 for gamma geq 1/2

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schr\"odinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$ on $L_2({\Bbb R})$. We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb R} V^{\gamma+1/2}(x)dx, (1) for the "limit" case $\gamma=1/2.$ This will imply improved estimates for the best constants $L_{\gamma,1}$ in (1), as $1/2<\gamma<3/2.Comment: AMS-LATEX, 15 page

Semiclassics in the lowest Landau band

This paper deals with the comparison between the strong Thomas-Fermi theory and the quantum mechanical ground state energy of a large atom confined to lowest Landau band wave functions. Using the tools of microlocal semiclassical spectral asymptotics we derive precise error estimates. The approach presented in this paper suggests the definition of a modified strong Thomas-Fermi functional, where the main modification consists in replacing the integration over the variables perpendicular to the magnetic field by an expansion in angular momentum eigenfunctions. The resulting DSTF theory is studied in detail in the second part of the paper.Comment: Latex2e, 31 page