Variation and series approach to the Thomas-Fermi equation

The Thomas-Fermi equation describing the screening of the Coulomb potential inside heavy neutral atoms is reconsidered. An accurate representation for its numerical solution was obtained by means of the variational principle. The proposed new solution has more precise asymptotic behaviour at large distances from the origin and allows us to obtain the exact value of the initial slope. The obtained new variational solution can also be developed in power series similar to the Baker’s ones but more precise even than some series solutions that have been recently obtained within the homotopy analysis method and a modified variational method.     

The Thomas-Fermi theory of atoms,molecules and solids

We place the Thomas-Fermi model of the quantum theory of atoms, molecules, and solids on a firm mathematical footing. Our results include: (1) A proof of existence and uniqueness of solutions of the nonlinear Thomas-Fermi equations as well as the fact that these solutions minimize the Thomas-Fermi energy functional, (2) a proof that in a suitable large nuclear charge limit, the quantum mechanical energy is asymptotic to the Thomas-Fermi energy, and (3) control of the thermodynamic limit of the Thomas-Fermi theory on a lattice.     

Eigenvalues of a nonlinear ground state in the Thomas-Fermi approximation

We study a nonlinear ground state of the Gross-Pitaevskii equation with a parabolic potential in the hydrodynamics limit often referred to as the Thomas-Fermi approximation. Existence of the energy minimizer has been known in literature for some time but it was only recently when the Thomas-Fermi approximation was rigorously justified. The spectrum of linearization of the Gross-Pitaevskii equation at the ground state consists of an unbounded sequence of positive eigenvalues. We analyze convergence of eigenvalues in the hydrodynamics limit. Convergence in norm of the resolvent operator is proved and the convergence rate is estimated. We also study asymptotic and numerical approximations of eigenfunctions and eigenvalues using Airy functions.     

Asymptotic properties of excited states in the Thomas-Fermi limit

Excited states are stationary localized solutions of the Gross-Pitaevskii equation with a harmonic potential and a repulsive nonlinear term that have zeros on a real axis. The existence and the asymptotic properties of excited states are considered in the semi-classical (Thomas-Fermi) limit. Using the method of Lyapunov-Schmidt reductions and the known properties of the ground state in the Thomas-Fermi limit, we show that the excited states can be approximated by a product of dark solitons (localized waves of the defocusing nonlinear Schrödinger equation with nonzero boundary conditions) and the ground state. The dark solitons are centered at the equilibrium points where a balance between the actions of the harmonic potential and the tail-to-tail interaction potential is achieved.     

An asymptotic analysis of positive solutions of generalized Thomas-Fermi differential equations — The sub-half-linear case

The existence and the asymptotics behavior for the large value of the variable of the positive solutions of generalized Thomas-Fermi equation presented in this article are proved. It is assumed that coefficient q(t) belongs to the class of regularly varying functions in the sense of Karamata. Properties of these functions and the Schauder-Tychonoff fixed point theorem are the main tools for the proofs.     

A ecessary and Sufficient Condition for The Stability of General molecular Systems

We study here the binding of atoms and molecules and the stability of general molecular systems including molecular ions. This is the first paper of a series devoted to the study of these general problems. We obtain here a general necessary and sufficient condition for the stability of general molecular ststem in the context of thomasz-Fermi-Von Weiasäcker, Thomas-Fermi-Dirac-Von Weizsaäcker, Hartree or Hartree-Fock theories

SUMARY OF PART 1

1.Introduction.

II.Presentation of the models

III.Diatomic molecular systems and hartree-Fock theory

IV.Diatomic molecular systems and Hartree or Thomas-Fermi theories

V.General molecular systems

Appendix 1: Hartree-Fock models when Z > N ― 1

Appendix 2: Dichotomy yields equal Lagrange multipliers

Appendix 3: The problem at infinty for the TRDW model     

Binding of atoms and stability of molecules in hartree and thomas-fermi type theories.

This paper is the third of a series devoted to the study of the binding of atoms, molecules and ions and of the stability of general molecular systems including molecular ions, in the context of Hartree and Thomas-Fermi type theories. For Thomas-Fermi-von WeizsÖcker or Thomas-Fermi-Dirac-von Weizsäcker models, we prove here that neutral systems can be bound and in view of the results shown in the preceding parts this yields the stability of arbitrary molecules (general neutral molecular systems). For the Hartree and Hartree-Fock models, we prove that neutral planar systems can be bound and this yields the stability of arbitrary tetraatomic molecules for instance. Various variants and extensions are also considered.     

The heat equation and reflected Brownian motion in time-dependent domains.: II. Singularities of solutions

We study the space-time Brownian motion and the heat equation in non-cylindrical domains. The paper is mostly devoted to singularities of the heat equation near rough points of the boundary. Two types of singularities are identified—heat atoms and heat singularities. A number of explicit geometric conditions are given for the existence of singularities. Other properties of the heat equation solutions are analyzed as well.     

Adaptive grid refinement for a model of two confined and interacting atoms

We have applied adaptive grid refinement to solve a two-dimensional Schrödinger equation in order to study the feasibility of a quantum computer based on extremely-cold neutral alkali-metal atoms. Qubits are implemented as motional states of an atom trapped in a single well of an optical lattice of counter-propagating laser beams. Quantum gates are constructed by bringing two atoms together in a single well leaving the interaction between the atoms to cause entanglement. For special geometries of the optical lattices and thus shape of the wells, quantifying the entanglement reduces to solving for selected eigenfunctions of a Schrödinger equation that contains a two-dimensional Laplacian, a trapping potential that describes the optical well, and a short-ranged interaction potential. The desired eigenfunctions correspond to eigenvalues that are deep in the interior of the spectrum where the trapping potential becomes significant. The spatial range of the interaction potential is three orders of magnitude smaller than the spatial range of the trapping potential, necessitating the use of adaptive grid refinement.     

On the leading correction of the Thomas-Fermi model: Lower bound

Summary We prove that the quantum mechanical ground state energy of an atom with nuclear chargeZ can be bounded from below by the sum of the Thomas-Fermi energy of the problem plusq/8Z ² plus terms of ordero(Z ² ).     

Binding of atoms and stability of molecules in hartree and thomas-fermi type theories.

This paper is the fourth of a series devoted to the study of the stability of general molecular systems in Thomas-Fermi or Hartree type models. In the preceding part, we proved the bindingof arbitrary neutral systems for Thomas-Fermi type theories and of planar neutral systems forthe Hartree model. In this part, we manage to get rid of this restriction and thus, prove thebinding and the stability of arbitrary neutral systems for the Hartree model.     

Rotating Two-Component Bose-Einstein Condensates

Recently, coupled systems of nonlinear Schr?dinger equations have been used extensively to describe Bose-Einstein condensates. In this paper, we study the structure of vortices for rotating two-component Bose-Einstein condensates (BEC) in a three-dimensional domain. We show that vortices is 1-rectifiable set, and give its mean curvature in the strong coupling (Thomas-Fermi) limit. In particular, we study effect of rotating term acting on the vortices.     

On a hydrogen atom

We represent the deterministic substitute of the Schrödinger equation for a hydrogen atom in precise accordance with the Bohm-de Broglie interpretation of quantum mechanics. To construct a mathematical model, the Weierstrass theorem and limit cycles are used. This method can be applied to many-electron atoms.     

The spatial behavior of rotating two-component Bose-Einstein condensates

In the understanding of the spatial behavior of interacting components of rotating two-component Bose-Einstein condensates, a central problem is to establish whether coexistence of all the components occurs, or the interspecies interaction leads to extinction, that is, configurations where one or more densities are null. In this paper, we prove that the interspecies interaction leads to extinction in the Thomas-Fermi approximation in dimension three.     

Single-electron Schrödinger equation for many-electron systems

We show that under certain conditions, an electron of a many-electron system can be described by the Schrödinger equation with a local Hamiltonian. The square root of the electron density plays the role of the wave function, and the interaction with other electrons is taken into account by averaging with the exact conditional probability. The equation dictates a redefinition of the ionization energy, which is tested with the examples of the hydrogen molecule and two-electron atoms.     

Statistical approach to the Jaynes-Cummings model

We study the dynamics of systems consisting of interacting two-level atoms and a field (microcavities). Such systems include the Jaynes-Cummings model. We formulate the problem and present a short history of it, derive a generalized kinetic equation for the system, find its solution, and show that this model allows describing photon emission and absorption.     

Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise

We consider a heat conduction model for solids. Nearest neighbour atoms interact as coupled oscillators exchanging velocities in such a way that the total energy is conserved. The system is considered under periodic boundary conditions. We will show that the system has a hydrodynamic limit given by the solution of the heat equation and we discuss some aspects of the model.     

Lie‐group method for solving the problem of fission product behavior in nuclear fuel

Calculation of the gas atom concentration is an important feature of all physical models of fission gas release. We apply Lie‐group method for determining symmetry reductions to the diffusion equation describing the fission gas release from nuclear fuel. The resulting nonlinear ordinary differential equation is solved numerically using nonlinear finite difference method. Effects of the dimensionless group constant, the time, and the grain radius on the concentration diffusion function have been studied, and the results are plotted. It is found that the concentration of gas atoms increases as the dimensionless group constant, the power index, and the time increase, and it decreases with increase of the grain radius. Copyright © 2013 John Wiley & Sons, Ltd.     

Sudden-perturbation approximation for the dirac equation

We solve the Dirac equation describing the behavior of a hydrogen-like atom interacting with a spatially inhomogeneous ultrashort electromagnetic field pulse in the sudden-perturbation approximation. We express the corresponding transition probabilities through the known inelastic atomic form factors widely used in the theory of relativistic collisions of charged particles with atoms.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 58–63, January, 2005.     

Binding of atoms and stability of molecules in hartree and thomas—fermi type theories

This paper is the sequel of a previous work where we showed a general necessary and sufficient condition for the stability of an arbitrary molecular system (possibly ionized) in the framework of Hartree or Thomas-Fermi type theories. This condition, roughly speaking, meant that certain particular subsystems have to be bound. We show here in particular that this condition reduces for general molecular system with nonnegative excess charge to the binding of all subsystems with the same property. For neutral inolecular systems, this reduces to the binding of all neutral subsystems. In both cases, all other subsystems can be bound. We also show that, for the Hartree-Fock and Hartree models, this condition involves only “physical” sulxystems We use these reduced conditions to conclude allout the stability or the binding in some particular cases. This work 1s also the second of a series devoted to these equations and we shall come back on the binding of neutral systems in Part 3.