一类Hamilton算子的重数和完备性及其应用

对于一类Hamilton算子,考虑其特征值的重数,以及特征向量组和根向量组的完备性.首先给出了特征值的几何重数、代数指标和代数重数,再结合特征向量和根向量的辛正交性得到了特征向量组和根向量组完备的充分必要条件,最后将上述结果应用于板弯曲方程、平面弹性问题和Stokes流等问题中.     

上三角无穷维Hamilton算子特征值的代数指标

无穷维Hamilton算子特征函数系是否完备与其代数指标有关,研究了上三角无穷维Hamilton算子特征值的代数指标问题,基于主对角元的特征值和特征向量的某些性质,得到上三角无穷维Hamilton算子的几何重数和代数重数.     

一类无穷维Hamilton算子根向量组的完备性

本文研究主对角元为常数的无穷维Hamilton算子的特征值问题.基于次对角元乘积的特征值和特征向量的某些性质,刻画此类Hamilton算子特征值分布、特征值的代数指标、特征向量(或一阶根向量)的辛正交关系及特征向量组和根向量组在辛Hilbert空间中完备的充要条件.     

一类四阶Hamilton算子特征值的代数指标

研究了一类四阶Hamilton算子H_A特征值的代数指标问题.根据算子A与Hamilton算子H_A的关系,讨论了Hamilton算子H_A特征值的几何重数,代数指标及代数重数.最后结合例子说明其结果的有效性.     

一类无界算子矩阵根子空间的完备性

本文研究了在弹性力学和线性Hamilton系统中出现的一类无界算子的性质,给出了这类算子的根向量组在Hilbert空间中Cauchy主值意义下完备的充分条件.最后,以板弯曲方程为例讨论了系统对应的算子根向量组的完备性并加以说明判别准则的有效性.     

基于二次算子族的无界Hamilton算子根向量组的基性质

利用无界Hamilton算子导出的二次算子族,本文研究了一类无界Hamilton算子根向量组的Schauder基性质.首先,建立了无界Hamilton算子的根向量与相应的二次算子族的根向量之间的关系.其次,借助二次算子族谱的相关性质,刻画了无界Hamilton算子的本征值分布以及本征值的代数指标,并得到了无界Hamilton算子的根向量组是某个Hilbert空间的一个块状Schauder基的充要条件.最后,将所得结果应用于矩形薄板弯曲问题.     

一类算子矩阵特征值的代数指标和代数重数

自伴算子特征值的几何重数与代数重数相等,但对于非自伴算子不一定成立,这主要是特征值的代数指标起着决定性的作用.讨论了一类非自伴算子矩阵特征值的几何重数,代数指标与代数重数.     

无穷维Hamilton 算子广义本征函数系的完备性及其在弹性力学中的应用

本文利用无穷维Hamilton 算子的结构特性, 得到由算子的基本本征函数和若当型本征函数构成的广义本征函数系在Cauchy 主值意义下完备的充分必要条件. 进而将结果应用于弹性力学中的板弯曲问题. 相应结论为Hamilton 体系下的分离变量法(弹性力学求解新体系) 提供了理论保证.     

一类具有转移条件的向量Sturm-Liouville问题的特征值

考虑了一类具有转移条件的向量Sturm-Liouville问题的特征值及其重数问题.首先构造了与问题相关的新内积和基本解,得到特征值的充要条件.在此基础上证明了二维情况下,问题特征值的代数重数与几何重数相等.     

两部件并联维修系统算子的性质

研究了两部件并联维修系统算子的性质,通过选取空间和定义算子将模型方程转化成了抽象柯西问题,证明了系统算子是定义域稠的预解正算子,0是系统算子的几何重数为1的本征值.讨论了系统算子的共轭算子及其定义域,证明了0是共轭算子的代数重数为1的特征值.     

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.     

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 H ₀ are considered. Next, some necessary and sufficient conditions for the completeness of the system of eigen or root vectors of H ₀ are obtained. Finally, the obtained results are tested in several examples.     

对边简支的矩形平面弹性问题的辛本征展开定理

对来源于平面弹性问题的Hamilton算子的本征值问题进行了研究．在矩形域内含位移和应力的混合边界条件下,首先求解了相应算子的本征函数．接着,证明了本征函数系的完备性,这为施行分离变量法求解相应问题提供了可行性．最后,利用文中的辛本征展开定理获得了问题的一般解．     

The algebraic multiplicity of the complex eigenvalues of population operator and its application

In this paper we discuss the algebraic multiplicity of the complex eigenvalue of population operator. Under certain condition we first prove that all the complex eigenvalues of this operator, except at most finitely many ones, are of algebraic multiplicity 1,and then, as an application of this result, we obtain the asymptotic expansion of the solution of corresponding population system.     

Hyperbolicity of periodic solutions of functional differential equations with several delays

We study conditions for the hyperbolicity of periodic solutions to nonlinear functional differential equations in terms of the eigenvalues of the monodromy operator. The eigenvalue problem for the monodromy operator is reduced to a boundary value problem for a system of ordinary differential equations with a spectral parameter. This makes it possible to construct a characteristic function. We prove that the zeros of this function coincide with the eigenvalues of the monodromy operator and, under certain additional conditions, the multiplicity of a zero of the characteristic function coincides with the algebraic multiplicity of the corresponding eigenvalue.     

板几何中具反射边界条件的迁移算子的谱分析

在Lp(1 p<∞)空间上研究了板几何中具反射边界条件下各向异性、连续能量、非均匀介质的迁移方程,证明了该迁移算子产生C0半群的Dyson-Phillips展开式的二阶余项在Lp(1相似文献   

A Nonself-Adjoint 1D Singular Hamiltonian System with an Eigenparameter in the Boundary Condition

In this paper, we study a nonself-adjoint singular 1D Hamiltonian (or Dirac type) system in the limit-circle case, with a spectral parameter in the boundary condition. Our approach depends on the use of the maximal dissipative operator whose spectral analysis is adequate for the boundary value problem. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations so that we can determine the scattering matrix of dilation. Moreover, we construct a functional model of the dissipative operator and specify its characteristic function using the solutions of the corresponding Hamiltonian system. Based on the results obtained by the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the dissipative operator and Hamiltonian system.     

人口算子的谱性质

本文讨论人口算子的谱性质,得到了人口算子本征值除可能有限个外均是代数单的,人口算子本征值新的分布性质,人口算子广义本征函数不构成基序列,Ｌ２［０,ｒ２］上的人口算子根子空间完整等．