壳模型哈密顿量本征值的美与奇

本征值问题是自然科学中基本运算之一，对于超大矩阵的对角化是当今许多科学问题的瓶颈。在应用原子核壳模型理论研究较重的原子核结构时，因为壳模型组态太大，通常的方法是基于各种物理考虑做某些组态截断，另一个思路是利用新的算法和飞速发展的计算机资源对这些大矩阵对角化或者近似对角化。总结了本课题组近年来在壳模型哈密顿量本征值近似方面研究的主要结果，包括最低本征值半经验公式及多种外推方法、本征值与对角元的相关性等。The eigenvalue problem is one of the fundamental issues of sciences. Many research fields have been challenged by diagonalizing huge matrices. The nuclear structure theorists face this problem in studies of medium-heavynuclei in terms of the nuclear shell model, in which the configuration space is too gigantic to handle. Thus one usually truncates the nuclear shell model configuration space based on various considerations. Another approach is to make use of super computers by various algorithms, and/or to obtain approximate eigenvalues. In this paper we review our recent efforts in obtaining approximate eigenvalues of the nuclear shell model Hamiltonian, with the focus on our semi-empirical approach and a number of extrapolation approaches towards predicting the lowest eigenvalue, as well as strong correlation between the sorted eigenvalues and the diagonal matrix elements, and so on.     

The convergence of the lowest eigenvalue of a real nearly-symmetric matrix

The lowest eigenvalue of a real nearly-symmetric matrix is expressed as a perturbation series in terms of the eigenvalues of the symmetric part and the matrix elements of the skew-symmetric part. It is shown that the resulting series is closely related to the perturbation series for the lowest eigenvalue of a related hermitian matrix. This enables the behaviour of the lowest eigenvalue of a nearly symmetric matrix as the dimension of the matrix is increased to be deduced from the behaviour of the lowest eigenvalue of a hermitian matrix. This is of considerable importance as the behaviour of the lowest eigenvalue of a hermitian matrix as the dimension of the matrix is increased can be much more readily established. A possible application to Boys' transcorrelated method of calculating atomic and molecular energies is suggested.     

New approach to synchronization in asymmetrically coupled networks

Global exponentially synchronization in asymmetrically coupled networks is investigated in this Letter. We extend eigenvalue based method to synchronization in symmetrically coupled network to synchronization in asymmetrically coupled network. A new stability criterion of eigenvalue based is derived. In this criterion, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of sum of column of asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability result is that it can analytical be applied to the asymmetrically coupled networks and overcome the complexity on calculating eigenvalues of coupling asymmetric matrix. Therefore, this condition is very convenient to use. Moreover, a necessary condition of this synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of coupling matrix.     

Application of h-adaptive,high order finite element method to solve radial Schrödinger equation

The h-adaptive, high order finite element method is applied to solve a second order one dimension eigenvalue problem. The finite element formulation for the Lobatto basis is given, for which basis functions of arbitrary order can be constructed. The adaptive algorithm is simple, yet very efficient and straightforward to implement. The algorithm is based on the observation that the expansion coefficients of Lobatto basis functions decay rapidly. It allows evaluating the smallest eigenvalues simultaneously with the comparable accuracy for all eigenvalues. The presented algorithm is applied to solve the radial Schrödinger equation with the Coulomb and the Woods–Saxon potentials. For both potentials the convergence rate is presented. After seven adaptive iterations nine-digit accuracy was obtained.     

Synchronizability on complex networks via pinning control

It is proved that the maximum eigenvalue sequence of the principal submatrices of coupling matrix is decreasing. The method of calculating the number of pinning nodes is given based on this theory. The findings reveal the relationship between the decreasing speed of maximum eigenvalue sequence of the principal submatrices for coupling matrix and the synchronizability on complex networks via pinning control. We discuss the synchronizability on some networks, such as scale-free networks and small-world networks. Numerical simulations show that different pinning strategies have different pinning synchronizability on the same complex network, and the consistence between the synchronizability with pinning control and one without pinning control in various complex networks.     

New eigenvalue based approach to synchronization in asymmetrically coupled networks

Locally and globally exponential stability of synchronization in asymmetrically nonlinear coupled networks and linear coupled networks are investigated in this paper, respectively. Some new synchronization stability criteria based on eigenvalues are derived. In these criteria, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of the sum of the column of the asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability results is that they can be analytically applied to the asymmetrically coupled networks and can overcome the complexity of calculating eigenvalues of the coupling asymmetric matrix. Therefore, these conditions are very convenient to use. Moreover, a necessary condition of globally exponential synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of the coupling matrix.     

Normal random matrix ensemble as a growth problem

In general or normal random matrix ensembles, the support of eigenvalues of large size matrices is a planar domain (or several domains) with a sharp boundary. This domain evolves under a change of parameters of the potential and of the size of matrices. The boundary of the support of eigenvalues is a real section of a complex curve. Algebro-geometrical properties of this curve encode physical properties of random matrix ensembles. This curve can be treated as a limit of a spectral curve which is canonically defined for models of finite matrices. We interpret the evolution of the eigenvalue distribution as a growth problem, and describe the growth in terms of evolution of the spectral curve. We discuss algebro-geometrical properties of the spectral curve and describe the wave functions (normalized characteristic polynomials) in terms of differentials on the curve. General formulae and emergence of the spectral curve are illustrated by three meaningful examples.     

Eigenvalues and their connection to transition rates for the Brownian motion in an inclined cosine potential

For certain parameters the motion of particles in an inclined cosine potential is bistable, i.e. particles are either in a locked or a running state. Fluctuations will cause transitions between these two states. First the connection of the transition rates with the lowest non-zero eigenvalue and the stationary solution of the Fokker-Planck equation is given. Then the eigenvalues of the Fokker-Planck equation for this Brownian motion problem are calculated using the matrix continued fraction method. Finally explicit results for these (generally complex) eigenvalues as a function of the averaged angle of inclination are shown for three typical friction constants and various temperatures.     

Radiative temperature dissipation in a finite atmosphere—I. The homogeneous case

We have developed a new approach toward solving problems of linear radiative relaxation of LTE temperature perturbations in a plane-parallel atmosphere of finite extent. We show that the mathematical problem is one of solving an integral eigenvalue equation, for which non-trivial solutions exist only for discrete values of the radiative relaxation time. The solutions for the spatial part of the perturbation constitute a complete and orthogonal set of basis functions, making it possible to solve more general problems of temperature relaxation. In applying this method to radiative relaxation in the middle atmosphere of earth, we show how the additional influences of photochemical coupling, advection by winds, and eddy diffusion by small-scale turbulence may be easily included using matrix perturbation techniques. We have solved the homogeneous integral equation for a wide variety of vertical thicknesses in an idealized homogeneous slab medium. Adopting a number of different analytic line profiles (rectangular, Doupler, Voigt, and Lorentz) we have obtained numerical solutions using an exponential-kernel method for solving the integral equation. The discrete eigenvalue “spectrum” is presented for vertical optical depths (0–10³) at line-center, and is used in solving several initial-value problems for a decaying temperature perturbation. We find that the eigenvalue spectrum is bounded from above by the lowest-order eigenvalue, and bounded from below by the familiar transparent approximation. The dependence of the lowest even eigenvalue on optical depth and the relative separation of the higher eigenvalues are found to depend sensitively on the line profile.     

自旋梯可积模型的研究

通过对自旋梯可积模型的研究,求出该模型的能量本征值和两体散射矩阵.用可积模型中的坐标Bethe Ansatz方法,首先由薛定谔方程求得能量的本征方程.设定波函数的具体形式,求出本征能量,然后利用能量本征方程和波函数的连续性求出两体散射矩阵.求出单粒子、双粒子和N₀个粒子的本征能量,同时求得粒子的两体散射矩阵.自旋梯可积模型的本征能量和两体散射矩阵可通过Bethe Ansatz的方法求得.     

Residue-squaring diagonalisation method and the anharmonic oscillator

A recently-formulated residue-squaring method for perturbation problems is subjected to an exacting test in its application to the problem of diagonalising the Hamiltonian of the nonlinear oscillator with quartic anharmonicity. Unlike other methods, this new iterative diagonalisation method enables several eigenvalues to be calculated simultaneously with little more labour than for a single eigenvalue. Values obtained for the four lowest even-parity levels of the anharmonic oscillator from just two or three iterations are shown to agree well with earlier accurate calculations. An approximate analytical formula for the energy levels is also presented.     

The Exceptional Jordan Eigenvalue Problem

We discuss the eigenvalue problem for 3 ×3 octonionic Hermitian matrices which is relevant to theJordan formulation of quantum mechanics. In contrast tothe eigenvalue problems considered in our previous work, all eigenvalues are real and solve theusual characteristic equation. We give an elementaryconstruction of the corresponding eigenmatrices, and wefurther speculate on a possible application to particle physics.     

Statistical properties of cross-correlation in the Korean stock market

We investigate the statistical properties of the cross-correlation matrix between individual stocks traded in the Korean stock market using the random matrix theory (RMT) and observe how these affect the portfolio weights in the Markowitz portfolio theory. We find that the distribution of the cross-correlation matrix is positively skewed and changes over time. We find that the eigenvalue distribution of original cross-correlation matrix deviates from the eigenvalues predicted by the RMT, and the largest eigenvalue is 52 times larger than the maximum value among the eigenvalues predicted by the RMT. The b₄₇₃\beta_{473} coefficient, which reflect the largest eigenvalue property, is 0.8, while one of the eigenvalues in the RMT is approximately zero. Notably, we show that the entropy function E(s)E(\sigma) with the portfolio risk σ for the original and filtered cross-correlation matrices are consistent with a power-law function, E(σ) ~ s^-g\sigma^{-\gamma}, with the exponent γ ~ 2.92 and those for Asian currency crisis decreases significantly.     

Eigenvalue density of cross-correlations in Sri Lankan financial market

We apply the universal properties with Gaussian orthogonal ensemble (GOE) of random matrices namely spectral properties, distribution of eigenvalues, eigenvalue spacing predicted by random matrix theory (RMT) to compare cross-correlation matrix estimators from emerging market data. The daily stock prices of the Sri Lankan All share price index and Milanka price index from August 2004 to March 2005 were analyzed. Most eigenvalues in the spectrum of the cross-correlation matrix of stock price changes agree with the universal predictions of RMT. We find that the cross-correlation matrix satisfies the universal properties of the GOE of real symmetric random matrices. The eigen distribution follows the RMT predictions in the bulk but there are some deviations at the large eigenvalues. The nearest-neighbor spacing and the next nearest-neighbor spacing of the eigenvalues were examined and found that they follow the universality of GOE. RMT with deterministic correlations found that each eigenvalue from deterministic correlations is observed at values, which are repelled from the bulk distribution.     

Accurate summation of the perturbation series for periodic eigenvalue problems

We show that algebraic approximants prove suitable for the summation of the perturbation series for the eigenvalues of periodic problems. Appropriate algebraic approximants constructed from the perturbation series for a given eigenvalue provide information about other eigenvalues connected with the chosen one by branch points in the complex plane. Such approximants also give those branch points with remarkable accuracy. We choose Mathieu's equation as illustrative example. Received 6 December 2000     

Effect of changing data size on eigenvalues in the Korean and Japanese stock markets

In this study, we attempted to determine how eigenvalues change, according to random matrix theory (RMT), in stock market data as the number of stocks comprising the correlation matrix changes. Specifically, we tested for changes in the eigenvalue properties as a function of the number and type of stocks in the correlation matrix. We determined that the value of the eigenvalue increases in proportion with the number of stocks. Furthermore, we noted that the largest eigenvalue maintains its identical properties, regardless of the number and type, whereas other eigenvalues evidence different features.     

A note on perfect revivals in finite waveguide arrays

We propose a simple and algorithmic method for designing finite waveguide arrays capable of diffractionless transmission of arbitrary discrete beams by virtue of perfect revivals. Our approach utilises an inverse matrix eigenvalue theorem published by Hochstadt in 1974, which states that the Jacobi matrix, describing the system's discrete evolution equations, is uniquely determined by its eigenvalues and the eigenvalues of its largest leading principal submatrix, as long as the two sets of eigenvalues interlace. It is further shown that, by arranging the two sets of eigenvalues symmetrically with respect to zero, the resulting Jacobi matrix has zero diagonal elements. Therefore, arrays with arbitrary revival lengths can be obtained by engineering only the inter-waveguide couplings.     

电子结构方法中带结构特征值问题的统一视角

在(相对论)电子结构方法中,四元数矩阵特征值问题和计算激发能的线性响应(Bethe-Salpeter)特征值问题是两个经常出现的结构特征值问题. 尽管前一个问题已被十分仔细地研究,后一个问题在一般形式下,即不假设电子Hessian正定性的复矩阵情况,并没有得到完全的理解. 鉴于它们非常相似的数学结构,本文从一个统一的角度研究了这两个问题,揭示了它们特征向量的“李群”结构,为将来设计对角化算法和数值优化方法提供了一个统一的框架. 利用和处理四元数矩阵特征值问题相同的归约算法,本文给出了表征线性响应问题特征值(实数、纯虚、或复数)的充分必要条件. 这一结果可以看作是实矩阵情况下已知条件的自然推广.     

Determination of eigenvalues of real symmetric para-p diagonal matrices

A method is presented for an accurate numerical determination of eigenvalues of real symmetric para-p diagonal matrices. The method takes advantage of the band structure to break up the matrix intop ×p blocks and performing algebraic operations including inversions on these blocks only, no matter what the size of the matrix is. The eigenvalues are determined independently one at a time. Thus any error in the determination of one eigenvalue does not affect the other eigenvalues. The method is ideally suited for the Schrödinger eigen alue problem of the anharmonic potentials, which is taken up in the following paper.     

紧束缚近似含时密度泛函理论的高效OpenMP并行化和GPU加速实现

紧束缚近似的含时密度泛函理论在多核和GPU系统下的高效加速实现,并应用于拥有成百上千原子体系的激发态电子结构计算．程序中采用了稀疏矩阵和OpenMP并行化来加速哈密顿矩阵的构建,而最为耗时的基态对角化部分通过双精度的GPU加速来实现．基态的GPU加速能够在保持计算精度的基础上达到8.73倍的加速比．激发态计算采用了基于Krylov子空间迭代算法,OpenMP并行化和GPU加速等方法对激发态计算的大规模TDDFT矩阵进行求解,从而得到本征值和本征矢,大大减少了迭代的次数和最终的求解时间．采用GPU对矩阵矢量相乘进行加速后的Krylov算法能够很快地达到收敛,使得相比于采用常规算法和CPU并行化的程序能够加速206倍．程序在一系列的小分子体系和大分子体系上的计算表明,相比基于第一性原理的CIS方法和含时密度泛函方法,程序能够花费很少的计算量取得合理而精确结果．