Hartree-Fock方程解集的结构

The maximal excess charge for a family of density-matrix-functional theories including Hartree-Fock and Müller theories

We will give a proof that the maximal excess charge for an atom described by a family of density-matrix-functionals, which includes Hartree-Fock and Müller theories, is bounded by an universal constant. We will use the new technique introduced by Frank et al (arXiv:1608.05625 [math-ph]).

On the Reduced Hartree-Fock Equations with a Small Anderson Type Background Charge Distribution

We demonstrate that the reduced Hartree-Fock equation (REHF) with a small Anderson type background charge distribution has an unique stationary solution by explicitly computing a screening mass at positive temperature.

Decay and Scattering in energy space for the solution of weakly coupled Schrödinger-Choquard and Hartree-Fock equations

We prove decay with respect to some Lebesgue norms for a class of Schrödinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space for the solutions to the systems of $N$ defocusing Schrödinger-Choquard equations with mass-energy intercritical nonlinearities in any space dimension and of defocusing Hartree-Fock equations, for any dimension $d\geq3$.

Boltzmann limit for a homogenous Fermi gas with dynamical Hartree-Fock interactions in a random medium

We study the dynamics of the thermal momentum distribution function for an interacting, homogenous Fermi gas on $\Z^3$ in the presence of an external weak static random potential, where the pair interactions between the fermions are modeled in dynamical Hartree-Fock theory. We determine the Boltzmann limits associated to different scaling regimes defined by the size of the random potential, and the strength of the fermion interactions.

Hartree-Fock方程的SCF序列收敛性

Convergence of SCF sequences for the Hartree-Fock equation

本文证明Hartree-Fock方程的SCF迭代序列在乘以酉矩阵后收敛，给出极限为方程解的充分条件，并证明密度算子收敛，从而确保SCF方法的有效性。

Weighted analyticity of Hartree-Fock eigenfunctions

We prove analytic-type estimates in weighted Sobolev spaces on the eigenfunctions of a class of elliptic and nonlinear eigenvalue problems with singular potentials, which includes the Hartree-Fock equations. Going beyond classical results on the analyticity of the wavefunctions away from the nuclei, we prove weighted estimates locally at each singular point, with precise control of the derivatives of all orders. Our estimates have far-reaching consequences for the approximation of the eigenfunctions of the problems considered, and they can be used to prove a priori estimates on the numerical s…

Minimizers for the Hartree-Fock-Bogoliubov Theory of Neutron Stars and White Dwarfs

We prove the existence of minimizers for Hartree-Fock-Bogoliubov (HFB) energy functionals with attractive two-body interactions given by Newtonian gravity. This class of HFB functionals serves as model problem for self-gravitating relativistic Fermi systems, which are found in neutron stars and white dwarfs. Furthermore, we derive some fundamental properties of HFB minimizers such as a decay estimate for the minimizing density. A decisive feature of the HFB model in gravitational physics is its failure of weak lower semicontinuity. This fact essentially complicates the analysis compared to the…

从Hartree和Hartree-Fock到Vlasov-Poisson方程的强半经典极限

Strong semiclassical limit from Hartree and Hartree-Fock to Vlasov-Poisson equation

本文研究了从Hartree和Hartree-Fock到Vlasov方程的半经典极限，在Schatten范数下证明了强拓扑中的显式界，并给出了费米子情形下交换项估计及收敛速率，适用于一般初始数据和任意固定时间区间。

On a Novel Effective Equation of the Reduced Hartree-Fock Theory

We show that there is an one-to-one correspondence between solutions to the Poisson-Landscape equations and the reduced Hartree-Fock equations in the semi-classical limit at low temperature. Moreover, we prove that the difference between the two corresponding solutions is small by providing explicit estimates.

Blow-up profile of neutron stars in the Hartree-Fock-Bogoliubov theory

We consider the gravitational collapse for neutron stars in the Hartree-Fock-Bogoliubov theory. We prove that when the number particle becomes large and the gravitational constant is small such that the attractive interaction strength approaches the Chandrasekhar limit mass slowly, the minimizers develop a universal blow-up profile. It is given by the Lane-Emden solution.

Stability of the Hartree-Fock model with temperature

This paper is devoted to the Hartree-Fock model with temperature in the euclidean space. For large classes of free energy functionals, minimizers are obtained as long as the total charge of the system does not exceed a threshold which depends on the temperature. The usual Hartree-Fock model is recovered in the zero temperature limit. An orbital stability result for the Cauchy problem is deduced from the variational approach.

The Hartree and Hartree-Fock equations in Lebesgue $L^p$ and Fourier-Lebesgue $\hat{L}^p$ spaces

We establish some local and global well-posedness for Hartree-Fock equations of $N$ particles (HFP) with Cauchy data in Lebesgue spaces $L^p \cap L^2 $ for $1\leq p \leq \infty$. Similar results are proven for fractional HFP in Fourier-Lebesgue spaces $ \hat{L}^p \cap L^2 \ (1\leq p \leq \infty).$ On the other hand, we show that the Cauchy problem for HFP is ill-posed if we simply work in $\hat{L}^p \ (2<p\leq \infty).$ Analogue results hold for reduced HFP. In the process, we prove the boundedeness of various trilinear estimates for Hartree type non linearity in these spaces which may be of i…

Decay in energy space for the solution of fourth-order Hartree-Fock equations with general non-local interactions

We prove the decay in the energy space for the solution to the defocusing biharmonic Hartree-Fock equations with mass-supercritical and energy-subcritical Choquard-type nonlinearity in space dimension $d\geq3$. We treat both the free and the perturbed by a potential case. As a direct consequence, we obtain large-data scattering in $H^2(\mathbb R^d)^N, \, N\geq 1$.

The Hartree-Fock equations in modulation spaces

We establish both a local and a global well-posedness theories for the nonlinear Hartree-Fock equations and its reduced analog in the setting of the modulation spaces on $\mathbb R^d$. In addition, we prove similar results when a harmonic potential is added to the equations. In the process, we prove the boundedeness of certain multilinear operators on products of the modulation spaces which may be of independent interest.

一维Hartree-Fock方程的散射

Scattering for the one dimensional Hartree Fock equation

本文研究一维Hartree-Fock方程在小局部化初值和有限测度势下的散射行为，利用时空共振框架证明直接项与交换项的非线性抵消消除了长程散射，从而得到向线性波的散射。

Hartree-Fock type systems: existence of ground states and asymptotic behavior

In this paper we consider an Hartree-Fock type system made by two Schrödinger equations in presence of a Coulomb interacting term and a cooperative pure power and subcritical nonlinearity, driven by a suitable parameter $β\geq 0$. We show the existence of semitrivial and vectorial ground states solutions depending on the parameters involved. The asymptotic behavior with respect to the parameter $β$ of these solutions is also studied.