含Delta势薛定谔方程的数值方法研究

发布时间：2018-03-10 13:46

本文选题：定态薛定谔方程　切入点：动力学非线性薛定谔方程　出处：《南京师范大学》2016年博士论文　论文类型：学位论文

【摘要】：薛定谔方程是量子物理中的基本方程,它不仅在物理领域有很多应用,而且在数学领域也备受关注。带分布势如δ(x)、δ'(x)及其线性组合的薛定谔方程,是描述玻色或费米气体的一类重要模型,也可以用来模拟半导体异质结构或奇异外势作用下超冷稀薄原子气体的凝聚问题,等等。本文主要研究含这类分布势的定态和动力学薛定谔方程的数值方法,主要成果如下：第一部分集中在含分布势定态薛定谔方程束缚态问题的界面方法研究。首先,我们考虑含单δ(x)或对称双δ(x)势阱的薛定谔方程,利用显式跳跃浸入界面方法和Peskin浸入边界方法对方程进行离散,得到了相应的广义和标准代数特征值问题,并证明了前者可以转化为标准代数特征值问题。进而,利用带位移的反幂法和QR方法,对原方程束缚态问题进行了多方面的数值研究。其次,对于含-αδ(x)+bδ'(x)势且在原点带质量跳跃的薛定谔方程,我们分别考察了其浸入界面方法和显式跳跃浸入界面方法离散,得到了相应的广义和标准代数特征值问题,并针对这些离散问题的适定性与求解等方面给出了相关的理论分析。由显式跳跃浸入界面方法离散得到的标准特征值问题,可以用标准方法进行求解。然而,由浸入界面方法得到的是一个“隐式”代数特征值问题,我们设计了相容的不动点迭代与带位移的反幂法相结合的算法,实现了相关束缚态能级与波函数的数值计算。理论和数值研究表明,对于这两类薛定谔方程束缚态特征值问题,Peskin的浸入边界方法、显式跳跃浸入界面方法和浸入界面方法都是有效的、稳定的以及收敛的,且对于能级的计算精度可以达到方法的精度。第二部分针对凝聚态物理中一个重要模型即含δ(x)势场的动力学非线性薛定谔方程,研究其多辛几何结构以及相关的多辛几何算法。由于Delta外势的出现,此方程不能像无外势那样写成多辛哈密尔顿系统。通过一些泛函设定,我们提出了此模型的“弱”多辛哈密尔顿形式描述,并从理论上研究了相关“弱”意义下的一些局部和整体守恒律。我们指出,由于空间平移不变性被打破,此系统的动量不再守恒。进而,我们构造了新的Runge-Kutta和Runge-Kutta-Nystrom方法,讨论了它们的离散多辛性,并证明了这些多辛几何算法严格保持原系统的归一化守恒律。为了数值比较,我们还利用界面方法构造了非多辛离散格式的数值算例。这些数值研究表明,多辛几何算法的优势在于对归一化守恒律的精确保持,并且界面点是否落在等分节点上对计算结果影响不大。对于能量守恒律,多辛Runge-Kutta-Nystrom算法的保持精度远高于非多辛算法。不管是多辛算法还是非多辛算法,可以将能量、宇称等诸多守恒性质在一定精度下长时间稳定保持。
[Abstract]:Schrodinger equation is a basic equation in quantum physics. It has many applications in the field of physics and has attracted much attention in the field of mathematics. It is an important model for describing Boson or Fermi gas. It can also be used to simulate the condensation of ultra-cold rarefied atomic gas under the action of semiconductor heterostructure or singular external potential. The main results are as follows: the first part focuses on the interfacial method for the bound state problem of the stationary Schrodinger equation with the distributed potential. In this paper, we consider the Schrodinger equation with single 未 X) or symmetric double 未 X) potential well. The explicit jump immersion interface method and Peskin immersion boundary method are used to discretize the equation, and the corresponding generalized and standard algebraic eigenvalue problems are obtained. It is proved that the former can be transformed into a standard algebraic eigenvalue problem. Furthermore, by using the inverse power method with displacement and the QR method, the bound state problem of the original equation is numerically studied in many aspects. For the Schrodinger equation with a potential of-伪未 X) and a mass jump at the origin, we investigate the discretization of the immersion interface method and the explicit jump immersion interface method, and obtain the corresponding generalized and standard algebraic eigenvalue problems. The relevant theoretical analysis on the suitability and solution of these discrete problems is given. The standard eigenvalue problems obtained from the discrete method of explicit jump immersion interface can be solved by the standard method. An implicit algebraic eigenvalue problem is obtained from the immersion interface method. We design a consistent fixed point iteration algorithm combined with the inverse power method with displacement. The theoretical and numerical studies show that Peskin's immersion boundary method is used to solve the eigenvalue problems of the bound states of the Schrodinger equation. The explicit jump immersion method and the immersion interface method are both effective, stable and convergent. In the second part, an important model in condensed matter physics, that is, the dynamical nonlinear Schrodinger equation with 未 X) potential field, can be obtained. The multi-symplectic geometry structure and the related multi-symplectic geometry algorithm are studied. Because of the appearance of Delta external potential, the equation can not be written as a multi-symplectic Hamiltonian system like no external potential. In this paper, we present a "weak" multi-symplectic Hamiltonian formal description of the model, and theoretically study some local and global conservation laws in the sense of "weak". We point out that the spatial translation invariance is broken. The momentum of the system is no longer conserved. Furthermore, we construct new Runge-Kutta and Runge-Kutta-Nystrom methods, discuss their discrete multi-symplectic properties, and prove that these multi-symplectic geometric algorithms strictly maintain the normalized conservation laws of the original system. Numerical examples of non-polysymplectic discrete schemes are also constructed by using the interface method. These numerical studies show that the advantages of multi-symplectic geometric algorithms lie in the exact preservation of normalized conservation laws. For energy conservation law, the accuracy of multi-symplectic Runge-Kutta-Nystrom algorithm is much higher than that of non-polysymplectic algorithm. Whether it is multi-symplectic algorithm or non-polysymplectic algorithm, the energy can be obtained by multi-symplectic algorithm or non-polysymplectic algorithm. Many conservation properties such as parity are kept stable for a long time under certain precision.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.8

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