算子函数的Weyl定理的判定

Weyl定理反映了算子特征值的分布特点.运用新的谱集,给出了有界线性算子满足Weyl定理的新的判定方法.进一步通过该谱集,刻画了算子函数满足Weyl定理的充要条件.     

A-函数关于Weyl函数m的反表示

考虑Simon反谱理论新方法中引入的A-函数,根据Weyl函数m关于A-函数的表示关系,利用广义函数和Fourier变换的方法求出A-函数关于Weyl函数m的反表示,该结论表明A-函数的本质是广义函数.     

关于厄米特多项式的新微分公式及其在量子光学中的应用

基于量子光学厄米特多项式和Weyl对应规则,该文给出了一类双变量厄米特多项式的生成函数.考虑到Weyl编序的相似变换不变性特征,还得到了另一个厄米特多项式广义生成函数,这些生成函数能被用于研究量子光场的非经典特征.     

算子函数的Weyl定理及其稳定性

设H是无限维复Hilbert空间,B(H)表示H上的有界线性算子全体构成的集合.本文对B(H)中使得f(T)满足Weyl定理的算子进行刻画,其中f是T的谱集的某个邻域上的解析函数.同时,也对算子函数的Weyl定理及算子Weyl定理的摄动之间的关系进行了讨论.     

奇异Sturm-Liouville问题Weyl函数比较定理及应用

在正常S-L问题比较定理基础上,给出分离型边界条件下右定奇异S-L问题当势函数不同时Weyl函数比较定理,同时结合Weyl函数性质得到右定S-L问题特征值比较定理,且进一步给出势函数不同时奇异左定S-L问题特征值比较定理.     

实轴上集序列的强渐近行为

首先基于点序列渐进分布的概念(分布函数是连续函数),提出实轴上集序列强渐近分布的概念(分布函数是绝对连续函数),是模1一致分布概念的推广.其次给出强渐近分布的Weyl型准则,改进Nakajima和Ohkubo的结果.最后利用强渐近分布的Weyl型准则研究一类实轴上迭代函数系统轨迹的强渐近行为.     

单边算子的多线性交换子在Triebel-Lizorkin空间上的加权有界性

该文在单边意义下采用权的外推法研究了Calderón-Zygmund奇异积分算子,离散面积函数,Weyl分数次积分与Lipschitz函数生成的多线性交换子从加权Lebesgue空间到加权Triebel-Lizorkin空间上的有界性.     

一类Weyi型单李超代数

本文研究了单李超代数的构造理论.借助于张量积方法,定义了一类Weyl型结合超代数和一类Weyl型李超代数,并且证明了这类Weyl型结合超代数和Weyl型李超代数是单的充分必要条件.     

加权的Coxeter群C_n的左胞腔(英文)

仿射Weyl群(C_n,S)可以看做仿射Weyl群(A_m,S_m)(其中m∈{2n-1,2n,2n+1})在其某个群自同构α下的固定点集合.A_m上的长度函数l_m可以看作C_n上的一个权函数.因此通过对仿射Weyl群(A_m,S_m)在其群自同构α下的固定点集合的研究可以得出加权的Coxeter群(C_n,l_m)的性质.本文给出了加权的Coxeter群(C_n,l_(2n))对应于划分42~(n-2)1的所有胞腔的清晰刻画.     

一类Weyl型单李超代数

本文研究了单李超代数的构造理论．借助于张量积方法，定义了一类Weyl型结合超代数和一类Weyl型李超代数，并且证明了这类Weyl型结合超代数和Weyl型李超代数是单的充分必要条件．     

Skew-Self-Adjoint Dirac System with a Rectangular Matrix Potential: Weyl Theory, Direct and Inverse Problems

A non-classical Weyl theory is developed for skew-self-adjoint Dirac systems with rectangular matrix potentials. The notion of the Weyl function is introduced and direct and inverse problems are solved. A Borg?CMarchenko type uniqueness result and the evolution of the Weyl function for the corresponding focusing nonlinear Schr?dinger equation are also derived.     

Nonlinear Schrödinger equation in a semi-strip: Evolution of the Weyl–Titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions

Rectangular matrix solutions of the defocusing nonlinear Schrödinger equation (dNLS) are studied in quarter-plane and semi-strip. Evolution of the corresponding Weyl–Titchmarsh (Weyl) function is described in terms of the initial Weyl function and boundary conditions. In the next step, the initial Weyl function is recovered (for the quarter-plane case) from the long-time asymptotics of the wave function considered at the boundary. Thus, it is shown that the evolution of the Weyl function is uniquely defined by the boundary conditions. Moreover, a procedure to recover solutions of dNLS (uniquely defined by the boundary conditions) is given. In a somewhat different way, the same boundary value problem is also dealt with in a semi-strip (for the case of a quasi-analytic initial condition).     

Weyl functions of generalized dirac systems: Integral representation,the inverse problem and discrete interpolation

We study self-adjoint Dirac systems and subclasses of canonical systems (which generalize Dirac systems) and obtain explicit and global solutions for direct and inverse problems. We also derive a local Borg-Marchenko-type theorem, integral representation of the Weyl function, and results on the interpolation of Weyl functions.     

Sine-Gordon theory in a semi-strip

Initial-boundary value problems for sine-Gordon and complex sine-Gordon equations in a semi-strip are treated. The evolution of the Weyl function and a uniqueness result are obtained for the complex sine-Gordon equation. The evolution of the Weyl function as well as an existence result and a procedure to recover solution are given for the sine-Gordon equation. It is shown that for a wide class of examples the solutions of the sine-Gordon equation are unbounded in the quarter-plane.     

Riemann-Cartan-Weyl geometries, quantum diffusions and the equivalence of the free maxwell and Dirac-Hestenes equations

We introduce the Riemann-Cartan-Weyl (RCW) space-time geometries of quantum mechanics with the most general trace-torsion non-exact Weyl 1-form, and characterize it in the Clifford bundle. Two electromagnetic potentials appear in the Weyl form, one having a zero field and the other one being the codifferential of a 2-form. We give the derivation of the non-linear equation for the wave function producing the exact Weyl one-form, which also defines the amplitude of a Dirac-Hestenes spinor operator field (DHSOF). We prove an equivalence between the free Maxwell equation for an extremal electromagnetic field and the Dirac-Hestenes equation for a DHSOF on a Riemann-Cartan-Weyl manifold, associating the electromagnetic potentials of the Weyl one-form with the internal electromagnetic potentials derived from the rotational dependance of a DHSOF. We show that this association produces a breaking of detailed balance in the spin plane. We discuss the relations with stochastic electrodynamics and the Navier-Stokes equation.     

An evolution equation in phase space and the Weyl correspondence

The Weyl correspondence that associates a quantum-mechanical operator to a Hamiltonian function on phase space is defined for all tempered distributions on R². The resulting Weyl operators are shown to include most Schroedinger operators for a system with one degree of freedom. For each tempered distribution, an evolution equation in phase space is defined that is formally equivalent to the dynamics of the Heisenberg picture. The evolution equation is studied both through a separation of variables technique that expresses the evolution operator as the difference of two Weyl operators and through the geometric properties of the distribution. For real tempered distributions with compact support the evolution equation has a unique solution if and only if the Weyl equation does. The evolution operator has skew-adjoint extensions that solve the evolution equation if the distribution satisfies an orthogonal symmetry condition.     

Commutation methods for Schrödinger operators with strongly singular potentials

We explore the connections between singular Weyl–Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schrödinger operators (also known as Bessel operators). We also investigate the connections with the generalized Bäcklund–Darboux transformation.     

Semiseparable Integral Operators and Explicit Solution of an Inverse Problem for a Skew-Self-Adjoint Dirac-Type System

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form f(l) exp{-2ilD}{\phi(\lambda)\,{\rm exp}\{-2i{\lambda}D\}}, where f{\phi} is a strictly proper rational matrix function and D = D* ≥ 0 is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case.     

On a formula of the generalized resolvents of a nondensely defined Hermitian operator

The Weyl function and the prohibited lineal, corresponding to a given space of boundary values of a nondensely defined Hermitian operator, are introduced and investigated. The prohibited lineal is characterized in terms of the limiting values of the Weyl function. An analogue of M. G. Krein's formula for the resolvent is obtained and its connection with the space of boundary values is found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1658–1688, December, 1992.