Exact Radial Solution of the Non-relativistic Schr?dinger Equation for the Helium Atom with the Potential Harmonic Method

We proposed a simple potential harmonic(PH) scheme for calculating the non-relativistic radial correlation energies of atomic systems.The scheme was applied to the low-lying n1S(n=1,2) and n3S(n=2,3) states of the helium atom.The results exhibit a very stable convergence characterization in both the angular and radial directions with PH and generalized Laguerre functions(GLF) respectively,even though the method is non-variational one.The ninth significant figure of the non-relativistic radial energy(NRE) calculated for the ground state exactly agrees with that of the most accurate literature data from the modified configuration interaction method.The convergent NRE′s for the excited states 21S,23S and 33S with the similar accuracy were also obtained.     

Deformed soliton,breather,and rogue wave solutions of an inhomogeneous nonlinear Schrdinger equation

We use the 1-fold Darboux transformation (DT) of an inhomogeneous nonlinear Schro¨dinger equation (INLSE) to construct the deformed-soliton, breather, and rogue wave solutions explicitly. Furthermore, the obtained first-order deformed rogue wave solution, which is derived from the deformed breather solution through the Taylor expansion, is different from the known rogue wave solution of the nonlinear Schro¨dinger equation (NLSE). The effect of inhomogeneity is fully reflected in the variable height of the deformed soliton and the curved background of the deformed breather and rogue wave. By suitably adjusting the physical parameter, we show that a desired shape of the rogue wave can be generated. In particular, the newly constructed rogue wave can be reduced to the corresponding rogue wave of the nonlinear Schro¨dinger equation under a suitable parametric condition.     

Applications of Tauberian theorems in some problems in mathematical physics

We give some multidimensional Tauberian theorems for generalized functions and show examples of their application in mathematical physics. In particular, we consider the problems of stabilizing the solutions of the Cauchy problem for the heat kernel equation, multicomponent gas diffusion, and the asymptotic Cauchy problem for a free Schrödinger equation in the norms of different Banach spaces among others.     

Matlab package for the Schrödinger equation

The paper presents a Matlab package for the linear time-(in)dependent Schr?dinger equation, based on the Hermite spectral method. The matrix form of the discretized problem is suitable for the linear algebra capabilities of Matlab. The high accuracy and efficiency of the algorithm is proved by many examples taken from the literature.      

Continuity properties of the integrated density of states on manifolds

We first analyze the integrated density of states (IDS) of periodic Schrödinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with both continuous and discontinuous IDS. Subsequently, alloy-type perturbations of the periodic operator are considered. The randomness may enter both via the potential and the metric. A Wegner estimate is proven which implies the continuity of the corresponding IDS. This gives an example of a discontinuous “periodic” IDS which is regularized by a random perturbation.     

2-Cocycles of original deformative Schr?dinger-Virasoro algebras

Both original and twisted Schr?dinger-Virasoro algebras, and also their deformations were introduced and investigated in a series of papers by Henkel, Roger and Unterberger. In the present paper we aim at determining the 2-cocycles of original deformative Schr?dinger-Virasoro algebras. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10471091, 10671027), Foundation of Shanghai Education Committee (Grant No. 06FZ029) and “One Hundred Talents Program” from University of Science and Technology of China     

Bound states of a system of two fermions on a one-dimensional lattice

We consider the Hamiltonian of a system of two fermions on a one-dimensional integer lattice. We prove that the number of bound states N(k) is a nondecreasing function of the total quasimomentum of the system k ∈ [0, π]. We describe the set of discontinuity points of N(k) and evaluate the jump N(k +0) − N(k) at the discontinuity points. We establish that the bound-state energy z _n (k) increases as the total quasimomentum k ∈ [0, π] increases. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 47–57, April, 2006.     

快速数值差分递推改进算法在超宽连续谱模拟中的应用研究

对快速数值差分递推公式进行了改进,使之能够求解带有脉冲自陡峭项和脉冲内喇曼散射项的非线性薛定谔方程.通过与传统孤子解析结果及分步傅里叶方法数值结果的对比分析表明,快速数值差分递推改进算法是一种快速而准确的数值计算方法,它不仅能够同步考虑光学媒质中的群速色散作用和非线性克尔作用,而且将所有高阶非线性项对光脉冲传输的影响也...     

立方五次方非线性Schrdinger方程的动力学性质研究

采用辛算法数值求解了一维立方五次方非线性Schrdinger方程,研究了不同非线性参数下非线性Schrdinger方程的动力学性质.数值结果表明,随着立方非线性参数的增加,系统经历了拟周期状态、混沌状态和周期状态,且在五次方项的调制下,呼吸子解可以退化为单孤子解.