非等谱广义耦合非线性Schrdinger方程的局部化孤立波(英文)

利用Hirota双线性方法求解了一个非等谱广义耦合非线性Schr(o|¨)dinger方程,得到它的N-孤子解.其中单孤子可以描述一个任意大振幅且具有时间和空间双重局部性的孤立波,这种特征与所谓的"怪波"相一致.此外,借助于图像描述了二孤子的相互作用.     

Exact Controllability for the Fourth Order Schr¨odinger Equation

The boundary controllability of the fourth order Schr(o)dinger equation in a bounded domain is studied.By means of an L2-Neumann boundary control,the authors prove that the solution is exactly controll...     

Hardy spaces H_L~p(R~n) associated with operators satisfying k-Davies-Gaffney estimates

Let L be a one-to-one operator of type ω having a bounded H∞ functional calculus and satisfying the k-Davies-Gaffney estimates with k ∈ N. In this paper, the authors introduce the Hardy space HLp(Rn) with p ∈ (0, 1] associated with L in terms of square functions defined via {e-t2kL}t>0 and establish their molecular and generalized square function characterizations. Typical examples of such operators include the 2k-order divergence form homogeneous elliptic operator L1 with complex bounded measurable coefficients and the 2k-order Schrdinger type operator L2 := (-Δ)k + Vk, where Δ is the Laplacian and 0≤V∈Llock(Rn). Moreover, as an application, for i ∈ {1, 2}, the authors prove that the associated Riesz transform ▽k(Li-1/2) is bounded from HLip (Rn) to Hp(Rn) for p ∈ (n/(n + k), 1] and establish the Riesz transform characterizations of HL1p (Rn) for p ∈ (rn/(n + kr), 1] if {e-tL1 }t>0 satisfies the Lr-L2 k-off-diagonal estimates with r ∈ (1, 2]. These results when k := 1 and L := L1 are known.     

带色散项的高阶非线性Schrdinger方程的精确解

对一类带色散项的高阶非线性Schrdinger方程的精确解进行研究.通过行波约化,将一类带色散项的高阶非线性Schrdinger方程化为一个高阶非线性常微分方程.再借助于计算机代数系统Mathematica通过构造非线性常微分方程的精确解,成功获得了一系列含有多个参数的包络型精确解,当精确解中参数取特殊值时可以得到两种新型的复合孤子解.并讨论了这两种孤子解存在的参数条件.     

Heat Kernel on Analytic Subvariety

In this paper, the author extends Peter Li and Tian Gang’s results on the heat kernel from projective varieties to analytic varieties. The author gets an upper bound of the heat kernel on analytic varieties and proves several properties. Moreover, the results are extended to vector bundles. The author also gets an upper bound of the heat operators of some Schr¨ondinger type operators on vector bundles. As a corollary, an upper bound of the trace of the heat operators is obtained.     

A version of Gordon's theorem for multi-dimensional Schrödinger operators

We consider discrete Schrödinger operators in with , and study the eigenvalue problem for these operators. It is shown that the point spectrum is empty if the potential is sufficiently well approximated by periodic potentials. This criterion is applied to quasiperiodic and to so-called Fibonacci-type superlattices.

     

一类Schrǒdinger算子的谱分析

本文研究了Hilbert空间L^2（R^2）上由势函数V（x）（V≥0,连续）给出的一类Schrǒdinger算子H=-△＋V的谱。本文的主要结果：（1）H的谱σ（H）不会出现本性谱与离散谱交替出现的情况,其谱要么是离散的,要么从infσcos（H）开始全是本性谱;（2）lim‖x‖→∞V（x）=∞是σcos（H）=φ的充要条件。（3）借助于讨论H的Zhis-lin谱,在一定的条件下。lim‖x‖→∞V（x）=0是σcos（H）=[∞,0）的充要条件。我们还提出了几个没有解决的问题。     

Strichartz estimates for the Schrödinger equation with radial data

We prove an endpoint Strichartz estimate for radial solutions of the two-dimensional Schrödinger equation:

     

Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials

We prove a dispersive estimate for the time-independent Schr?dinger operator H  =   − Δ  + V in three dimensions. The potential V(x) is assumed to lie in the intersection

Schrödinger uncertainty relation for a quantum oscillator in a thermostat

We introduce the Schrödinger correlator as the holistic characteristic of two types of fluctuation correlations in quantum dynamics and in statistical thermodynamics. We are the first to derive it using methods of thermofield dynamics for the coordinate-momentum variables of a quantum oscillator in a thermostat. We show that the obtained value ensures that the Schrödinger uncertainty relation becomes an equality at all temperatures. We find that the thermal equilibrium for the quantum oscillator has the sense of the thermal correlated coherent state and can be adequately described by a wave function with temperature-dependent amplitude and phase.