About poles of the resolvent, in a model for a harmonic oscillator coupled with massless scalar bosons
Absence of geometrical phases in the rotating stark effect
An upper and lower solution approach for a generalized Thomas-Fermi theory of neutral atoms.
Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential.
In [FS1] we announced a precise asymptotic formula for the ground-state energy of a non-relativistic atom. The purpose of this paper is to establish an elementary inequality that plays a crucial role in our proof of that formula. The inequality concerns the Thomas-Fermi potentialVTF = -y(ar) / r, a > 0, where y(r) is defined as the solution of⎧ y''(x) = x-1/2y3/2(x),⎨ y(0) = 1,⎩ y(∞) = 0.
Atomistic to Continuum limits for computational materials science
The present article is an overview of some mathematical results, which provide elements of rigorous basis for some multiscale computations in materials science. The emphasis is laid upon atomistic to continuum limits for crystalline materials. Various mathematical approaches are addressed. The setting is stationary. The relation to existing techniques used in the engineering literature is investigated.
Auger spectra and different ionic charges following and sub-shells photoionization of Kr atoms.
Bounds on the excess charge and the ionization energy for the Hellmann-Weizsacker model
Diffusion Monte Carlo method: Numerical Analysis in a Simple Case
The Diffusion Monte Carlo method is devoted to the computation of electronic ground-state energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a fixed number of random walkers evolving according to a stochastic differential equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple one-dimensional example, we prove...
Excess charge for pseudo-relativistic atoms in Hartree-Fock theory.
Existence of minimizers for Kohn-Sham models in quantum chemistry
First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules
The ground-state energy and properties of any many-electron atom or molecule may be rigorously computed by variationally computing the two-electron reduced density matrix rather than the many-electron wavefunction. While early attempts fifty years ago to compute the ground-state 2-RDM directly were stymied because the 2-RDM must be constrained to represent an N-electron wavefunction, recent advances in theory and optimization have made direct computation of the 2-RDM possible. The constraints in...
Localization of the essential spectrum for relativistic -electron ions and atoms.
Macro- and microsimulations for a sublimation growth of SiC single crystals.
Multiscale Materials Modelling: Case Studies at the Atomistic and Electronic Structure Levels
Although the intellectual merits of computational modelling across various length and time scales are generally well accepted, good illustrative examples are often lacking. One way to begin appreciating the benefits of the multiscale approach is to first gain experience in probing complex physical phenomena at one scale at a time. Here we discuss materials modelling at two characteristic scales separately, the atomistic level where interactions are specified through classical potentials and the...
On recurrence relations for radial wave functions for the -th dimensional oscillators and hydrogenlike atoms: analytical and numerical study.
On the classical non-integrability of the Hamiltonian system for hydrogen atoms in crossed electric and magnetic fields
Hydrogen atoms placed in external fields serve as a paradigm of a strongly coupled multidimensional Hamiltonian system. This system has been already very extensively studied, using experimental measurements and a wealth of theoretical methods. In this work, we apply the Morales-Ramis theory of non-integrability of Hamiltonian systems to the case of the hydrogen atom in perpendicular (crossed) static electric and magnetic uniform fields.
On the current of large atoms in strong magnetic fields
In this talk I will discuss recent results on the magnetisation/current of large atoms in strong magnetic fields. It is known from the work (E. Lieb, J.P. Solovej, and J. Yngvason, “Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions”, Commun. Math. Phys. (1994), no. 161, 77-124) of Lieb, Solovej and Yngvason that the energy and density of atoms in strong magnetic fields are given to highest order by a Magnetic Thomas Fermi theory (MTF-theory) when the magnetic field strength...
On the essential spectrum of the Jansen-Hess operator for two-electron ions.
Perturbative treatment of the evolution operator associated with Raman couplings.
Semiclassics of the quantum current in very strong magnetic fields
We prove a formula for the current in an electron gas in a semiclassical limit corresponding to strong, constant, magnetic fields. Little regularity is assumed for the scalar potential . In particular, the result can be applied to the mean field from magnetic Thomas-Fermi theory . The proof is based on an estimate on the density of states in the second Landau band.