SYMMETRY OF POSITIVE SOLUTIONS FOR THE FRACTIONAL HARTREE EQUATION

In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.

ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA

In this paper,we study the decay estimate and scattering theory for the Klein-Gordon-Hartree equation with radial data in space dimension d≥3.By means of a compactness strategy and two Morawetz-type estimates which come from the linear and nonlinear parts of the equation,respectively,we obtain the corresponding theory for energy subcritical and critical cases.The exponent range of the decay estimates is extended to 0〈γ≤4 and γ〈d with Hartree potential V（x） =｜x｜-γ.

GEOMETRIC OPTICS FOR 3D-HARTREE-TYPE EQUATION WITH COULOMB POTENTIAL

This article considers a family of 3D-Hartree-type equation with Coulomb potential ｜x｜^-1, whose initial data oscillates so that a caustic appears. In the linear geometric optics case, by using the Lagrangian integrals, a uniform description of the solution outside the caustic, and near the caustic are obtained.

GLOBAL WELL-POSEDNESS AND BLOW-UP FOR THE HARTREE EQUATION

For 2 〈 y 〈 min{4, n}, we consider the focusing Hartree equation iut ＋ Au ＋ （｜x｜^-γ ＊ ｜u｜2）u = O, x∈ R^n Let M[u] and E[u] denote the mass and energy, respectively, of a solution u, and Q be the ground state of - △ ＋ Q = （｜x｜^-γ ＊ ｜Q｜^2）Q. Guo and Wang [Z. Angew. Math. Phy.,2014] established a dichotomy for scattering versus blow-up for the Cauchy problem of （0,1） if M[u]^l-ScE[u]^Sc 〈 M[Q] ^1-sc E[Q] ^（sc= r-2/2）. In this paper, we consider the complementary case M[u]^1-ScE[u]^sc 〉_ M[Q]^1-sc and obtain a criteria on blow-up and global existence for the Hartree equation （0.1）.

Electronic structure from equivalent differential equations of Hartree–Fock equations

A strict universal method of calculating the electronic structure of condensed matter from the Hartree-Fock equation is proposed.It is based on a partial differential equation(PDE)strictly equivalent to the Hartree-Fock equation,which is an integral-differential equation of fermion single-body wavefunctions.Although the maximum order of the differential operator in the Hartree-Fock equation is 2,the mathematical property of its integral kernel function can warrant the equation to be strictly equivalent to a 4 th-order nonlinear partial differential equation of fermion single-body wavefunctions.This allows the electronic structure calculation to eliminate empirical and random choices of the starting trial wavefunction(which is inevitable for achieving rapid convergence with respect to iterative times,in the iterative method of studying integral-differential equations),and strictly relates the electronic structure to the space boundary conditions of the singlebody wavefunction.

原子基态能量的高精度Roothaan-Hartree-Fock计算方法

量子力学中的Roothaan-Hartree-Fock(R-H-F)近似计算在原子、分子和材料电子结构计算中具有重要的意义,是其他高精度电子结构计算方法的基础.本工作根据开源网站Basis Set Exchange中丰富的基组资源,利用高斯型基组矩阵元具有解析表达式的特性,自行编写Roothaan-Hartree-Fock计算程序,可以较为方便的达到高精度的计算结果.同时本工作用不同大小的基组计算了多种原子和离子的基态能量,研究了He、Be、C2+等原子和离子在不同基组下的收敛特性.此外,基于自行编写的R-H-F程序计算得到He、Be、Ne的第一电离能,与实验对比最大误差不超过6.84%.

具弱耗散项的非线性Hartree方程

该文讨论了一类在量子理论中有着较多应用的具耗散项的非线性Hartree方程。分别从耗散系数和初值两个方面讨论了解的整体存在性条件。一方面,利用Strichartz估计,得到仅依赖于耗散系数的整体存在性条件。另一方面,也得到了仅依赖于初值大小的整体存在性条件。而且,还得到了一个整体解存在的小初值准则。

氦原子基态能量的Roothaan-Hartree-Fock计算

采用包含两个斯莱特基的"双ζ"函数说明了利用自洽场法求解基态氦原子Roothaan-Hartree-Fock方程的数值过程,计算得基态能量为-2.862 568 Hartree.利用基态的对称性,提出了通过求解泊松方程来计算库仑算符的方法,给出了交叠矩阵和单电子算符的矩阵元,并对自洽的标准作了讨论.

原子结构中求解Hartree-Fock方程的几种方法及其比较

主要介绍了原子结构理论中Hartree-Fock方程的几种近似求解方法,并对这几种方法的优缺点进行比较,总结出各种方法的适用范围.

一类具扰动项的Hartree方程驻波的强不稳定性(英文)

本文研究一类具扰动项的Hartree方程.通过建立交叉不变流研究其解整体存在的最佳门槛.进而,研究其解的强不稳定性.

四阶Wave-Hartree方程的Strichartz估计及其适应理论

研究含非局部非线性项的四阶波动方程,借助其Schrdinger结构的特征建立局部Strichartz估计,给出了该方程的局部适应性理论.进而,根据方程关于scaling变换是能量次临界的,并利用能量恒等式得到解的整体唯一性.

Klein-Gordon-Hartree方程驻波的不稳定性

研究如下Klein-Gordon-Hartree方程的驻波2u/t2-△u+ωu-(|x|-γ*|u|2)u=0,x∈RN.首先,通过构造合理的约束变分问题建立基态驻波解的存在性及其变分特征.然后,以变分特征为基础,证明存在一个Klein-Gordon-Hartree方程的解序列,其初始条件充分接近基态驻波,同时解在有限时间爆破,从而得到基态驻波的不稳定性.

具有Hartree类非线性项的非齐次薛定谔方程的初边值问题

用Banach空间的不动点定理及各种先验估计研究具有Hartree类非线性项的非齐次薛定谔方程,并且在给定的初值和边值条件下得到方程全局解的存在性结论,推广了相关文献的结果.

一类Schr?dinger-Hartree方程爆破解的门槛条件

该文在R^3中研究如下Schr?dinger-Hartree方程i?_tψ+△ψ=-(|x|^(-1)*|ψ|~α)|ψ|^(α-2)ψ,t>0,x∈R^3,α≥2.(P)利用Gagliardo-Nirenberg与方程(P)的质量守恒律,能量守恒律建立方程的发展不变流.以此为基础在7/3≤α<5时,得到其Cauchy问题的爆破解和整体解的门槛条件.

具次临界扰动项的非线性Hartree方程驻波的存在性

研究了一类具次临界扰动项的非线性Hartree方程驻波解的存在性.根据2个非线性项的特征,分2类情形建立相应的约束变分问题,得到了该类Hartree方程在2种情形下驻波的存在性.

INTERACTION MORAWETZ ESTIMATE AND A SIMPLIFIED PROOF ON THE ENERGY SCATTERING FOR HARTREE EQUAITONS

In this article, we give a simple proof on the energy scattering for the Hartree equations using the interaction Morawetz estimate that was originally introduced in [5].

带时间依赖的阻尼/增益分数阶Hartree方程解的全局存在性

利用Bootstrap论证方法,考虑一类带时间依赖的阻尼/增益分数阶Hartree方程解的全局存在性.分别从阻尼/增益系数和初值两方面讨论解全局存在的条件,得到了所讨论分数阶Hartree方程仅依赖于阻尼/增益系数,以及仅依赖于初值大小解的全局存在性条件.

A numerical Hartree self-consistent field calculation of an autoionization resonance parameters for a doubly excited 2s^2, 3s^2, and 4s^2 states of He atom with a complex absorbing potential

The self-consistent Hartree-Fock equation for the He atom is solved using the pseudospectral method. The Feshbach- type autoionization resonance parameters for doubly excited 2s2, 3s2, and 4s2 IS states of He have been determined by adding a complex absorbing potential to the Hamiltonian. The Riss-Meyer iterative and Pad6 extrapolation methods are applied to obtain reliable values for the autoionization resonance parameters, which are compared to previous results in the literature.

半空间上Hartree方程的Liouville型定理

该文研究半空间上的Hartree方程{−Δui(y)=∑_(j=1)^(n)∫∂R_(+)^(N)F(uj(x^(-),0))/|(x^(-),0)−y|^(N−α)dx^(-)g(ui(y)),y∈R_(+)^(N),∂ui/∂ν(x^(-),0)=∑_(j=1)^(N)∫R_(+)^(N)G(uj(y))/|(x^(-),0)−y|^(N−α)dyf(ui(x^(-),0)),(x^(-),0)∈∂R_(+)^(N)非平凡正解的非存在性结果.在对非线性项F,G,f,g作适当的假设下,证明上述方程只有常数的正解.该文用积分形式的移动平面法来证明.

Hartree方程的低正则算法

构建了一种Fourier积分器来求解非线性Hartree方程,这种指数型积分器是显式的,且可通过快速Fourier变换实现一阶收敛.通过严格的分析,证明对任意的γ>d/2,该格式对于H^(γ+1)空间中的任何初始数据都提供了一阶精度.即,固定时间T,存在常数C=C(T,‖u‖_L^(∞)([0,T];H^(γ+1)))>0,使得‖u^(n)-u(t_(n))‖_H^(γ)_((T)^(d))≤Cτ,其中u^(n)为在t_(n)=nτ处的数值解.