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．     

Two-fold completeness of root vectors of a system of quadratic pencils

We consider a spectral problem for a system of second order (in the spectral parameter) abstract pencils in a Hilbert space and prove the completeness and the Abel basis property of a system of eigenvectors and associated vectors. In some special cases, we obtain the expansion of vectors with respect to eigenvectors. Further, it is considered a relevant application of these abstract results to boundary-value problems for second and fourth order ordinary differential equations with a quadratic spectral parameter both in the equation and in boundary-value conditions.     

A class of nonlinear equations in Hilbert space and its application to completeness problems

Nonlinear equations arising in the spectral theory of self-adjoint operator functions and related completeness problems for eigenvectors are studied. A separation theorem about the values of the Rayleigh functional on solutions of a nonlinear equation is proved. This theorem is used, as a new approach to establish completeness of eigenvectors for some classes of self-adjoint operator functions. Examples from matrix pencils are given.     

Completeness of elementary solutions to a class of second order operator-differential equations

We present conditions of solvability of a boundary value problem for a class of second order operator-differential equations on a finite segment, study the behavior of the resolvent of the corresponding operator pencil, prove the double completeness of a system of the derived chains of eigenvectors and associated vectors corresponding to a boundary value problem on a segment, and establish the completeness of elementary solutions to the homogeneous equation in the solution space.     

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. I. Abstract theory

N. Dunford and J.T. Schwartz (1963) striking Hilbert space theory about completeness of a system of root vectors (generalized eigenvectors) of an unbounded operator has been generalized by J. Burgoyne (1995) to the Banach spaces framework. We use the Burgoyne's theorem and prove n-fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. The theory will allow to consider, in application, boundary value problems for ODEs and elliptic PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions.     

On small oscillations of a compressible stratified liquid

We study the structure of the spectrum and the completeness and basis property of a system of eigenvectors. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1614–1623, December, 2006.     

Eigenwaves in a lossy metal-dielectric waveguide

ABSTRACT

The problem of normal waves in a closed (shielded) regular waveguide of arbitrary cross-section is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operator-function on the complex plane is found. We also consider properties of system of eigenvectors and associated vectors of the operator-function. Double completeness of system of eigenvectors and associated vectors with a finite defect is established.     

Eigenwaves in waveguides with dielectric inclusions: completeness

We formulate the definition of eigenwaves and associated waves in a nonhomogeneously filled waveguide using the system of eigenvectors and associated vectors of a pencil and prove its double completeness with a finite defect or without a defect. Then, we prove the completeness of the system of transversal components of eigenwaves and associated waves as well as the ‘mnimality’ of this system and show that this system is generally not a Schauder basis. This work is a continuation of the paper Eigenwaves in waveguides with dielectric inclusions: spectrum. Appl. Anal. 2013. doi:10.1080/00036811.2013.778980 by Y. Smirnov and Y. Shestopalov. Therefore, we omit the problem statements and all necessary basic definitions given in the previous paper.     

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.     

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.     

Eigenvalues and eigenvectors of symmetric centrosymmetric matrices

It is proved that the eigenvectors of a symmetric centrosymmetric matrix of order N are either symmetric or skew symmetric, and that there are ?N/2? symmetric and ?N/2? skew symmetric eigenvectors. Some previously known but widely scattered facts about symmetric centrosymmetric matrices are presented for completeness. Special cases are considered, in particular tridiagonal matrices of both odd and even order, for which it is shown that the eigenvectors corresponding to the eigenvalues arranged in descending order are alternately symmetric and skew symmetric provided the eigenvalues are distinct.     

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.     

On the well-posedness of a boundary value problem for a class of fourth-order operator-differential equations

We find sufficient coefficient conditions for the well-posed solvability of a boundary value problem for a class of fourth-order operator-differential equations with multiple characteristics. Furthermore, we indicate the sharp values of norms of operators of intermediate derivatives in a Sobolev-type space. In addition, for the corresponding polynomial operator pencil, we prove the completeness of the part of its eigenvectors and associated vectors corresponding to the eigenvalues in the left half-plane.     

Spectral analysis of nonselfadjoint Schrödinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schrödinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrödinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrödinger boundary value problem are given.     

Spectral analysis of nonselfadjoint Schr(o)dinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schr(o)dinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator,and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schr(o)dinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schr(o)dinger boundary value problem are given.     

Closure of the squared Zakharov-Shabat eigenstates

By solving the inverse scattering problem for a third-order (degenerate) eigenvalue problem, we can find the closure of the squared eigenfunctions of the Zakharov-Shabat equations. The question of the completeness of squared eigenstates occurs in many aspects of “inverse scattering transforms” (solving nonlinear evolution equations exactly by inverse scattering techniques) as well as in various aspects of the inverse scattering problem. The method we use is quite suggestive as to how one might find the closure of the squared eigenfunctions of other eigenvalue equations, and we point the strong analogy between our results and the problem of finding the closure of the eigenvectors of a nonself-adjoint matrix.     

Spectral analysis of nonselfadjoint Schrdinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schrodinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrodinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrodinger boundary value problem are given.     

Linear operator pencilsA-λB with discrete spectrum

We study the spectral properties of non-self-adjoint linear pencilsA-B of bounded operators with discrete spectrum whereB is not necessarily bijective. The main results concern the minimality, completeness and basis properties of the corresponding eigenvectors and associated vectors.     

Modal analysis of piezoelectric bodies with voids. I. Mathematical approaches

The paper is concerned with the eigenvalue problems for piezoelectric bodies with voids in contact with massive rigid plane punches and coved by the system of open-circuited and short-circuited electrodes. The linear theory of piezoelectric materials with voids for porosity change properties according to Cowin–Nunziato model is used. The generalized statements for eigenvalue problem are obtained in the extended and reduced forms. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. The orthogonality relations for eigenvectors are obtained in different forms. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical, electric and “porous” boundary conditions and the moduli of piezoelectric solid with voids change, are established.