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

为离开斜的无限的维的 Hamiltonian 操作符，它有至多可计算的特征值，在 Cauchy 主要价值的意义完全的特徵函数系统的一个必要、足够的条件被使用介绍光谱对称和操作符的新直角的关系。而且，上述结果被扩大到一个更一般的盒子。最后，为从各向同性的飞机 magnetoelectroelastic 固体产生的操作员的特徵函数系统的完全性被描述说明标准的有效性。整个结果为一些力学方程在 Hamiltonian 系统为变量的分离提供理论保证。

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

<Abstract>The results of invertibility and spectrum for some different classes of infinite-dimensional Hamiltonian 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.

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.

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.

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.

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.

The Maximum Dissipative Extension of Schrodinger Operator

In the present paper we study the maximum dissipative extension of Schrodingeroperator.introduce the generalized indefinite metvic space and get the representation ofmaximum dissipative extension of Schrodinger operator in natural boundary space.make preparation for the further study longtime chaotic behaxior of infinite dimensiondynamics system in nonlinear Schrodinger equation.

Completeness in the sense of Cauchy principal value of the eigenfunction systems of infinite dimensional Hamiltonian operator

The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied, and the suffcient conditions of the completeness in the sense of Cauchy principal value of the eigenfunction systems of the infinite dimensional Hamiltonian operators are given. In the end, concrete examples are constructed to justify the effectiveness of the criterion.

Structure of the spectrum of infinite dimensional Hamiltonian operators

This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.

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.

Symmetry of the Point Spectrum of Infinite Dimensional Hamiltonian Operators and Its Applications

This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp（H） = σp （A） U σp1 （-A＊）. Using the characteristic of the set σp1（-A＊）, we divide the point spectrum σp （d） of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σp1（-A＊） and one part of σp（A） are symmetric with respect to the real axis each other. Based on this result, the symmetry of σp（H） is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.

Symmetry of the Point Spectrum of Upper Triangular Infinite Dimensional Hamiltonian Operators

In this paper,by using characterization of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H,a necessary and sufficient condition is obtained on the symmetry of σp(A) and σp1(-A*) with respect to the imaginary axis.Then the symmetry of the point spectrum of H is given,and several examples are presented to illustrate the results.

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

与简单地支持的二反面处于矩形的板的弯曲问题出现的无限维的 Hamiltonian 操作符的特徵函数系统被学习。起初，在 Cauchy 的主要价值的意义的扩大特徵函数系统的完全性被证明。然后，在一般感觉的扩大特徵函数系统的不完全性被证明。那么 symplectic 的完全性把方程弄弯的这种板的无限维的 Hamiltonian 操作符的直角的系统被证明。最后，无限的维的 Hamiltonian 系统的一般解决方案等价于解决方案函数系统系列扩大，因此它为这种方程基于 Hamiltonian 系统给变量的分离的方法的理论基础。

L^2×L^2中的一类无穷维Hamilton算子的剩余谱

该文得到了一类无穷维Hamilton算子的剩余谱和点谱存在的几个判别准则,从而给出 了求其剩余谱和点谱的方法．在此基础上构造了L2×L2中无穷维Hamilton算子的剩余谱 非空的具体例子,从而进一步验证了判别准则的有效性．

一些数学物理问题中的Hamilton方程

讨论了新的一系列在数学物理方程中微分方程的Hamilton正则表示 ,其中包括变系数 2阶对称方程的Hamilton系统 ,关于常系数的 4阶对称方程新的非齐次Hamilton表示 ,MKdV方程以及KP方程的正则表示·

一类无穷维Hamilton算子的谱分布

研究了一类有深刻力学背景的非自伴算子(即无穷维Hamilton算子)的谱,给出了一类无穷维Hamilton算子的谱的刻画.构造了一些具体的例子,把结果应用在波动方程生成的无穷维Hamilton算子上,得到了该算子的谱分布.

一类无穷维Hamilton算子的谱

研究了一类非自伴算子(无穷维Hamilton算子)的谱,刻画了一类无穷维Hamilton算子的点谱、剩余谱和连续谱,并举例验证了结果的有效性.

无穷维Hamilton算子特征值的代数指标

该文研究次对角元至少有一个为可逆的无穷维Hamilton算子特征值的代数指标问题.基于特征值和特征向量的某些性质,得到一类无穷维Hamilton算子的代数指标为1或2,并举例说明结果的有效性.

一类无穷维Hamilton算子族的特征函数系的完备性

对来源于波动方程中的一类无穷维Hamilton算子族,研究了其特征函数系的性质.得到如下结论:1)算子族中的每个算子的特征函数系存在一种新的正交关系,此种正交关系包含求解新体系中的辛正交关系;2)算子族中的每个算子的特征函数系在Cauchy主值意义下都是完备的,这为研究无穷维Hamilton算子补的特征函数系的完备性奠定了基础;3)得到波动方程更广泛的分离变量解.

弹性理论中上三角无穷维Hamilton算子根向量组的完备性

考虑弹性力学中一类上三角无穷维Hamilton算子.首先,给出此类Hamilton算子特征值的几何重数和代数指标,进而得到代数重数.其次,根据Hamilton算子特征值的代数重数确定其特征(根)向量组完备的形式,得到此类Hamilton算子特征(根)向量组的完备性是由内部算子特征向量组决定.最后,将所得结果应用到弹性力学问题中.