Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

For the off-diagonal infinite dimensional Hamiltonian operators, which have at most countable eigenvalues, a necessary and sufficient condition of the eigenfunction systems to be complete in the sense of Cauchy principal value is presented by using the spectral symmetry and new orthogonal relationship of the operators. Moreover, the above result is extended to a more general case. At last, the completeness of eigenfunction systems for the operators arising from the isotropic plane magnetoelectroelastic solids is described to illustrate the effectiveness of the criterion. The whole results offer theoretical guarantee for separation of variables in Hamiltonian system for some mechanics equations.

Invertibility of Infinite-Dimensional Hamiltonian Operators and Its Application to Plate Bending Equation

The results of invertibility and spectrum for some different classes of infinite-dimensional Hayniltonian operators, after a brief classification by domains. are given. By the above results, the associated infinite-dimensional Hamiltonian operator with simple supported rectangular plate is proved to be invertible. Furthermore, by a certain compactness, we find that the spectrum of this operator consists only of isolated eigenvalues with finite geometric multiplicity, which will play a significant role in finding the analytical and numerical solution based on Hamiltonian system for a class of plate bending equations.

Spectra of Off-diagonal Infinite-Dimensional Hamiltonian Operators and Their Applications to Plane Elasticity Problems

In the present paper, the spectrums of off-diagonal infinite-dimensional Hamiltonian operators are studied. At first, we prove that the spectrum, the continuous-spectrum, and the union of the point-spectrum and residual- spectrum of the operators are symmetric with respect to real axis and imaginary axis. Then for the purpose of reducing the dimension of the studied problems, the spectrums of the operators are expressed by the spectrums of the product of two self-adjoint operators in state spac,3. At last, the above-mentioned results are applied to plane elasticity problems, which shows the practicability of the results.

Spectral Description of a Class of Infinite-Dimensional Hamiltonian Operators and Its Application to Plane Elasticity Equations Without Body Force

In this paper,the results of spectral description and invertibility of upper triangle infinite-dimensionalHamiltonian operators with a diagonal domain are given.By the above results,it is proved that the infinite-dimensionalHamiltonian operator associated with plane elasticity equations without the body force is invertible,and the spectrumof which is non-empty and is a subset of R.

Completeness of the System of Root Vectors of 2×2 Upper Triangular Infinite-Dimensional Hamiltonian Operators in Symplectic Spaces and Applications

The authors investigate the completeness of the system of eigen or root vectors of the 2×2 upper triangular infinite-dimensional Hamiltonian operator H 0.First,the geometrical multiplicity and the algebraic index of the eigenvalue of H0 are considered.Next,some necessary and sufficient conditions for the completeness of the system of eigen or root vectors of H0 are obtained.Finally,the obtained results are tested in several examples.

An algorithm and its application for obtaining some kind of infinite-dimensional Hamiltonian canonical formulation

Using factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical ＇σ/σx＇-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.

Completeness of system of root vectors of upper triangular infinitedimensional Hamiltonian operators appearing in elasticity theory

This paper deals with a class of upper triangular infinite-dimensional Hamilto- nian operators appearing in the elasticity theory. The geometric multiplicity and algebraic index of the eigenvalue are investigated. Furthermore, the algebraic multiplicity of the eigenvalue is obtained. Based on these properties, the concrete completeness formulation of the system of eigenvectors or root vectors of the Hamiltonian operator is proposed. It is shown that the completeness is determined by the system of eigenvectors of the operator entries. Finally, the applications of the results to some problems in the elasticity theory are presented.

Eigenvalue problem of a class of fourth-order Hamiltonian operators

The eigenvalue problem of a class of fourth-order Hamiltonian operators is studied. We first obtain the geometric multiplicity, the algebraic index and the algebraic multiplicity of each eigenvalue of the Hamiltonian operators. Then, some necessary and sufficient conditions for the completeness of the eigen or root vector system of the Hamiltonian operators are given, which is characterized by that of the vector system consisting of the first components of all eigenvectors. Moreover, the results are applied to the plate bending problem.

On the ascent of infinite dimensional Hamiltonian operators

In this paper, the ascent of 2 × 2 infinite dimensional Hamiltonian operators and a class of 4 × 4 infinite dimensional Hamiltonian operators are studied, and the conditions under which the ascent of 2 × 2 infinite dimensional Hamiltonian operator is 1 and the ascent of a class of 4 × 4 infinite dimensional Hamiltonian operators that arises in study of elasticity is2 are obtained. Concrete examples are given to illustrate the effectiveness of criterions.

Bosonic Operator Realization of Hamiltonian for a Superconducting QuantumInterference Device

Based on the appropriate bosonic phase operator diagonalized in the entangled state representation we construct the Hamiltonian operator model for a superconducting quantum interference device. The current operator and voltage operator equations are derived.

非厄密系统哈密顿量的本征值和本征态

利用扭结哈密顿算子构造了非厄密哈密顿量,探讨了非厄密哈密顿量本征值和本征态的性质.结果表明:非厄密哈密顿量的能量本征值为复数,且随着模型参数和角度的变化而变化,可能会出现奇异点;通过理论推导可确定本征值奇异点的位置和个数.非厄密哈密顿量本征矢量之间的正交归一性呈现与传统量子力学完全不同的特点;考虑基尔霍夫电流定律,用电阻、电感及电容构造电路,可获得非厄密哈密顿量.

基于近端算子PHMC的机载雷达高度表参数估计

传统雷达高度表参数估计算法在面对参数的高维特性时会出现过拟合情况,导致参数估计精确度降低。为此,提出一种新颖的基于近端算子修正的哈密顿蒙特卡洛(PHMC)算法,通过统计学的手段估计高程参数。首先假设高程参数具有稀疏特性,并使用拉普拉斯分布对其进行表征,这种稀疏先验可表征高程突变的地形场景。稀疏先验与似然函数之间为非共轭关系,使用分层贝叶斯的方法获得后验分布函数的闭合解,采用哈密顿蒙特卡洛(HMC)方法通过采样的方式解决贝叶斯推论中的参数估计问题,引入近端算子提供次梯度完成参数估计。仿真数据验证了所提PHMC算法优于其他传统算法。

势函数V(r)=Z^2r^(-4)-d_1d_2r^(-3)的离子的Hamiltonian本征方程的精确解

采用连分法,得到离子之间相互作用势为V(r)=Z2r-4-d1d2r-3的离子的Hamiltonian算符的精确能量本征值和能量本征函数.

无界分块算子矩阵的可分解性及其应用

无界分块算子矩阵广泛地出现于系统理论、非线性分析以及发展方程问题等领域,在理论和实际应用两方面都受到广泛关注。首先,利用算子局部谱理论得到无界分块算子矩阵可分解性的刻画,其次,给出算子矩阵可分解性保持对角稳定的条件,推广并得到分块算子矩阵在无界情形下的一些局部谱性质。最后,作为应用考察Hamilton算子的可分解性并举例予以说明。

曲面上构造三次3-连通非Hamiltonian地图的一种方法(英文)

Tutte在1946年构造性证明了并非每个简单的3-凸胞腔都是Hamiltonian的后,人们又陆续提出了多种构造三次3-连通非Hamiltonian平面图的方法,但无一能用于在一般曲面上寻找三次3-连通非Hamiltonian地图.本文提出了一种新的构造方法,可在任一个曲面上构造出三次3-连通非Hamiltonian地图.

一类新的孤子族、可积耦合及其Hamiltonian结构

基于零曲率方程及实李代数so(3,R),建立了一类新的孤子方程族,并通过创建新的loop代数的方法构建了该孤子族的可积耦合族,然后利用变分恒等式得到了与之相对应的Hamiltonian结构。

广义协变Hamilton系统与结构算子的应用

研究了完整广义Hamilton系统平衡下的微分方程,得到了几何势函数的对数解,运用结构算子重新表示了由几何括号的刚性定理推导的引理公式,得到了一个推论,并举例说明。

Completeness of eigenfunction systems for the product of two symmetric operator matrices and its application in elasticity

The completeness theorem of the eigenfunction systems for the product of two 2 × 2 symmetric operator matrices is proved. The result is applied to 4 × 4 infinite-dimensional Hamiltonian operators. A modified method of separation of variables is proposed for a separable Hamiltonian system. As an application of the theorem, the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained. Finally, a numerical test is analysed to show the correctness of the results.

On Feasibility of Variable Separation Method Based on Hamiltonian System for a Class of Plate Bending Equations

The eigenfunction system of infinite-dimensional Hamiltonian operators appearing in the bending problem of rectangular plate with two opposites simply supported is studied. At first, the completeness of the extended eigenfunction system in the sense of Cauchy＇s principal value is proved. Then the incompleteness of the extended eigenfunction system in general sense is proved. So the completeness of the symplectic orthogonal system of the infinite-dimensional Hamiltonian operator of this kind of plate bending equation is proved. At last the general solution of the infinite dimensional Hamiltonian system is equivalent to the solution function system series expansion, so it gives to theoretical basis of the methods of separation of variables based on Hamiltonian system for this kind of equations.

势函数V(r)=A_1r^(-6)+A_2r^(-4)+A_3r^2的Hamiltonian本征方程的精确解

本文采用连分法得到相互作用势为V(r)=A1r-6 +A2r-4 +A3r2 的Hamiltonian算子的精确的能量本征值和能量本征函数.