积分-微分算子的迹公式

该文研究积分-微分算子的迹公式,它在反问题、特征值的数值计算、可积系统理论等有着重要的应用.得到了Dirichlet-Robin边界条件和Dirichlet边界条件下积分-微分算子的迹公式.     

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。     

关于奇异模迹的同余与整除恒等式

Zagier研究发现奇异模的迹是一些权为3/2的模形式的傅里叶系数. 结果进而人们研究发现了这些奇异模在各种条件下的迹公式和同余性质. 近来, Ahlgen利用模形式在Hecke算子作用下的关系式证明了奇异模迹的一个精确关系式. 基于此,我们研究得到了一些关于奇异模的迹和Hurwitz-Kronecker类数有趣的同余和整数恒等式.     

对称幺半范畴与相对Lefschetz-Verdier迹公式

本文通过一系列具体的实例展示对称么半范畴中对偶和迹的重要作用,并介绍其在平展上同调中的新应用—万有局部零调性的刻画和相对Lefschetz-Verdier迹公式.     

四元数半空间上模形式的维数公式 *

利用Selberg迹公式导出次数为 2的四元数半空间上尖点形式的一个维数公式 ,并计算出一些共轭类对维数公式的贡献 ,同时还得到模群共轭类划分的一些结果 .     

与含有三个位势矩阵谱问题相联系的非线性发展方程族及其Hamilton结构

从含有三个位势的4×4矩阵谱问题出发,导出两类非线性发展方程.然后利用迹公式,给出了这两类方程的广义Hamilton结构.     

带磁场的一维Schr?dinger方程的逆散射问题

本文研究带磁场的一维Schr?dinger方程的逆散射问题.利用其散射矩阵的有关性质以及函数变换的方法,获得位势p可以被散射系数决定的结果,并推广了迹公式的成立范围.     

带磁场的一维Schr\"{o}dinger方程的逆散射问题

本文研究带磁场的一维Schr\"{o}dinger方程的逆散射问题. 利用其散射矩阵的有关性质以及函数变换的方法, 获得位势$p$可以被散射系数决定的结果, 并推广了迹公式的成立范围.     

基于李代数sl(m+1,R)的多分量扰动AKNS孤子梯队

基于李代数sl(m+1,R),提出了一个新的多分量矩阵谱问题,进而利用零曲率公式构造了新的多分量扰动AKNS孤子梯队.利用迹恒等式构造了双哈密顿结构,同时给出了遗传递推算子.     

一类整函数零点的渐近公式

借助Rouché定理、留数定理及渐近分析的方法,给出了整函数f(z)=zmsinz-α(0≠α∈R,m∈Z+)零点的渐近公式及渐近迹.这种方法也适用于其它整函数的零点估计.     

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.     

Hill-Type Formula and Krein-Type Trace Formula for Hamiltonian Systems

In this paper, we give a survey on the Hill-type formula and its applications.Moreover, we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions, which include the standard Neumann, Dirichlet and periodic boundary conditions. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Further, based on the Hill-type formula, we derive the Krein-type trace formula. As applications, we give nontrivial estimations for the eigenvalue problem and the relative Morse index.     

Discrete Sturm-Liouville Problems With Parameter in the Boundary Conditions

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

Existence of a spectral measure for second-order delta dynamic equations on semi-infinite time scale intervals

In this study, we prove existence of a spectral measure (or orthogonality measure) for second-order delta dynamic equations on semi-infinite time scale intervals. A Parseval equality and an expansion in eigenfunctions formula are established in terms of the spectral measure. The result obtained unifies the well-known results on existence of a spectral measure for Sturm-Liouville operators on the real semi-axis and for semi-infinite Jacobi matrices, and extends them to variety of numerous time scales which may, in particular, be fractals.     

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.     

Existence and multiplicity of weak solutions for a class of fractional Sturm-Liouville boundary value problems with impulsive conditions

In this paper, we consider the existence and multiplicity of weak solutions for a class of fractional differential equations with non-homogeneous Sturm-Liouville conditions and impulsive conditions by using the critical point theory. In addition, at the end of this paper, we also give the existence results of infinite weak solutions of fractional differential equations under homogeneous Sturm-Liouville boundary value conditions. Finally, several examples are given to illustrate our main results.     

Inverse problem for Sturm-Liouville and hill equations

Summary We discuss the inverse Sturm-Liouville problem on a finite interval by the method of transformation kernel. The -function, the Fredholm determinant of the transformation kernel, is explicitly written down in terms of the spectral data, from which a very explicit representation formula for the potential is deduced, and well-posedness of the inverse problem is established. The above method is also applicated to the inverse problem for Hill equations, in particular to the isospectral problem. We obtain an analog of FIT formula and a regularity theorem.     

Matrix Finite-Zone Dirac-Type Equations

Weyl-Titchmarsh matrix functions play an essential role in the spectral theory of Dirac-type equations (Oper. Theory: Adv. Appl.107 (1999)). In this paper, we have constructed a class of Weyl-Titchmarsh matrix-functions generating potentials of finite-zone type. It has turned out that the corresponding potentials have derivatives of an arbitrary order. Using the above-mentioned results, we deduce the matrix analogue of the trace formula for finite-zone matrix potentials. In the last part of the paper, we consider separately the scalar case of Dirac-type equations. For this case, we have constructed finite-zone potentials in explicit forms and proved that these potentials are quasiperiodical. We note that for scalar Schrödinger equations the corresponding results are well known (see Invent. Math.30 (1975), 217-274; “Soliton and the Inverse Scattering Transform,” SIAM, Philadelphia, 1981; Rev. Sci. Technol.23 (1983), 20-50; “Theory of Solitons, The Method of Inverse Problem,” New York, 1984; “Inverse Sturm-Liouville Problems,” VSP, Zeist, 1987; “Inverse Spectral Theory,” Academic Press, New York, 1987).     

非线性算子方程变号解的存在性及其应用

本文利用锥理论讨论了非线性算子方程变号解的存在性,并将抽象结果应用于Sturm-Liouville两点边值问题。所得结果无论在理论上还是在应用上都是新的.     

一类四阶奇异半正Sturm-Liouville边值问题的正解

在Sturm-Liouville边界条件下研究较广泛的一类四阶奇异半正微分方程,得到其C2[0,1]正解与C3[0,1]正解存在的新结果,并给出了其正解与该边值问题的格林函数之间的某些联系.