项目名称: 薛定谔类型方程解的弱奇异性研究
项目编号: No.11201142
项目类型: 青年科学基金项目
立项/批准年度: 2013
项目学科: 数理科学和化学
项目作者: 毛仕宽
作者单位: 华北电力大学
项目金额: 22万元
中文摘要: Schrodinger方程是量子力学及数学物理中的一个基本方程,它的地位类似于经典力学中牛顿方程的地位。对它的研究在数学物理和微分方程中有着广阔的应用前景。本项目主要研究具有变系数和谐振子势的Schrodinger方程以及与之相关的具有变系数和磁场的Schrodinger方程。此时,方程的解构成酉交算子半群。我们首先研究了当变系数和势函数是自由谐振子(相应地,自由均匀磁场算子)的扰动的时候,解在非共鸣时间点(即,非周期时间点)上的弱奇异性(即,Sobolev空间意义下的微局部奇异性)的传播问题。然后,利用非共鸣时间点上的情况给出了共鸣时间点(即,周期时间点)上解的弱奇异性的传播、消失、新奇异性的产生等结果。
中文关键词: 薛定谔方程;传播;奇异性;谐振子;磁场
英文摘要: As one of the fundmental equations in quantum mechanics and mathematical physics, Schrodinger equation plays a similar role as the Newton equation in classical mechanics. Studying for it has a lot of applications in mathematical physics and differential equations. We mainly consider the Schrodinger equations with variable coefficients and harmonic potentials, and also the related Schrodinger equations with variable coefficients and magnetic fields. In this case, the solutions can be represented by a unitary semigroup. We first studied the propagation of weak singularities (i.e., microlocal singularities in the sense of Sobolev spaces) at non-resonant time for solutions to the Schrodinger equations with perturbed harmonic oscillators (respectively, with perturbed constant magnetic fields). Then, we give results of propagation, dispersion and creation of new singularities at resonant time in terms of the results at non-resonant time.
英文关键词: Schrodinger equation;propagation;singularity;harmonic oscillator;magnetic field