N维向量Sturm-Liouville算子的正则迹及其在反谱问题中的应用

研究了赋予一般分离型边界条件的N维向量Sturm-Liouville方程的特征值问题,获得了该系统的一个正则迹公式.迹公式不仅形式美观,而且它在反谱理论中具有重要的作用.     

Inversion of Trace Formulas for a Sturm-Liouville Operator

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.     

The Norm and Regularized Trace of the Cauchy Transform

In this paper, the norm of the Cauchy transform C is obtained on the space L ²(D, dμ), where dμ = ω(|z|) dA(z). Also, (for the case ω ≡ 1), the first regularized trace of the operator C* C on L ²(Ω) is obtained. The results are illustrated by examples, with different specific choices of the function ω and the domain Ω.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 844–853.Original Russian Text Copyright ©2005 by M. R. Dostanic.     

On the Similarity of Some Differential Operators to Self-Adjoint Ones

The paper is devoted to the study of the similarity to self-adjoint operators of operators of the form , in the space with weight . As is well known, the answer to this problem in the case is positive; it was obtained by using delicate methods of the theory of Hilbert spaces with indefinite metric. The use of a general similarity criterion in combination with methods of perturbation theory for differential operators allows us to generalize this result to a much wider class of weight functions .     

一类算子迹的不等式

将一个算子迹的不等式进行加细,并获得了一个新的算子迹不等式.     

偶阶非对称微分算子离散谱准则

本文研究了由2n阶复系数J-对称微分算式生成的J-自伴微分算子谱的离散性,分别得到了一类J-自伴微分算子谱离散的充分条件与必要条件,为判断一类微分算子谱的离散性提供了若干准则．     

关于算子迹的Bellman不等式

本文将矩阵中关于半正定 Hermite矩阵的 Bellman不等tr(AB) k tr(Ak Bk) ,k =1 ,2 ,…推广到 Hilbert空间 ,得到关于正的迹算子的相应不等式     

Regularized traces of a perturbed laplace-beltrami operator on the unit two-dimensional sphere

The paper contains formulas for regularized traces of the Laplace-Beltrami operator on the unit sphere under perturbation determined by a bounded complex-valued potential and a sufficiently complete justification of these formulas. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 702–705, May, 2000.     

一类奇型Sturm-Liouville算子的逆问题

研究了奇型Sturm-Liouville算子的逆问题.对于固定的n∈N,证明了Sturm-Liouville问题(1.3)-(1.5)的第n个特征值λ_n(q,H)关于H是严格单调增加的,及一组不同边界条件下的第n个特征值的谱集合{λ_n(q,H_k)}_(k=1)~(+∞)能够唯一确定(0,πr)上的势函数q(x).     

A trace formula of a boundary value problem for the operator Sturm-Liouville equation

We obtain a regularized trace formula for the operator Sturm-Liouville equation with a boundary condition depending on a spectral parameter.     

一类积—微分算子的离散本征值

本文讨论的积-微分算子是一类以众多应用领域为背景的无界、非自伴线性算子.在较一般的假设下,籍助L~2空间的线性算子理论,我们证明了这类算子存在离散本征值的充分条件,并获得了可供实际工作者参考的估计式.     

Location of zeros for Neumann Sturm-Liouville eigenfunctions

By means of variational method, we give some observations about the location of zeros of Neumann Sturm-Liouville eigenfunctions.     

一个特征值问题的迹公式

本文利用逼近理论的方法,分别提出了一个特征值问题在周期条件下以及在无穷衰减条件下的迹公式。     

关于矩阵和算子迹的一组不等式

给出关于半正定矩阵迹和正算子迹的一组不等式,得到与邱贤忠关于实数的不等式的类似结果.     

Perturbations of existence families for abstract Cauchy problems

In this paper, we establish Desch-Schappacher type multiplicative and additive perturbation theorems for existence families for arbitrary order abstract Cauchy problems in a Banach space: ; . As a consequence, we obtain such perturbation results for regularized semigroups and regularized cosine operator functions. An example is also given to illustrate possible applications.

     

Szego-Type Limit Theorems for Generalized Discrete Convolution Operators

We study the asymptotic behavior of the averaged f-trace of a truncated generalized multidimensional discrete convolution operator as the truncation domain expands. By definition, the averaged f-trace of a finite-dimensional operator A is equal to , where n is the dimension of the space in which the operator A acts, the set of numbers γ_k, k = 1,..., n, is the complete collection of eigenvalues of the operator A, counting multiplicity; a generalized discrete convolution is an operator from the closure of the algebra generated by discrete convolution operators and by operators of multiplication by functions admitting a continuous continuation onto the sphere at infinity.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 265–277.Original Russian Text Copyright © 2005 by I. B. Simonenko.     

非自伴Dirac算子的迹公式

本文研究了非自伴Dirac算子的一般两点边值问题的渐近迹,首先运用平移算子得到了其Cauchy问题解的渐近式,并由此及边界条件,构造了整函数ω(λ),利用它将边界条件分为八种基本类型,最后采用留数的方法,得到了四种主要类型的特征值的渐近迹公式。     

Hilbert空间中算子迹的Bellman不等式

本文利用矩阵论和泛函分析知识,证明了在Hilbert空间中算子迹的Bellman不等式tr(AB)^k≤tr(A^kB^k)当A,B为正迹类算子时,对一切自然数k都成立;当A,B为自伴迹类算子时,对一切偶数k都成立.