有限区间上带复权因子的微分算子逆谱问题的全局解

Inverse problems with a general transfer condition

We consider a Sturm-Liouville operator on a finite interval as well as a scattering problem on the real line both with transfer conditions at the origin. On a finite interval we show that the the Titchmarsh-Weyl $m$-function can be uniquely determined from two spectra for the same equation but with varied boundary conditions at one end of the interval. In addition, we prove that the $m$-function can also be uniquely reconstructed from one spectrum and the corresponding norming constants. For the scattering problem on the real line we assume that the potential has compact essential support. For…

Inverse scattering problems where the potential is not absolutely continuous on the known interior subinterval

The inverse scattering problem for the Schr$\mathrm{\ddot{o}}$dinger operators on the line is considered when the potential is real valued and integrable and has a finite first moment. It is shown that the potential on the line is uniquely determined by the left (or right) reflection coefficient alone provided that the potential is known on a finite interval and it is not absolutely continuous on this known interval.

谱在区间、有限区间系或正长康托尔集上的Jacobi矩阵的逆散射问题

On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a finite system of intervals or a Cantor set of positive length

对势为快减的Sturm-Liouville算子求解逆散射问题，得到了反射系数$s_\pm$和可逆算子$I+H_{s_\pm}$，其中$ H_{s_\pm}$是与符号$s_\pm$相关的Hankel算子. Marchenko-Fadeev定理（在连续情况下）和Guseinov定理（在离散情况下）保证了逆散射问题解的唯一性。在这篇文章中，我们提出了以下自然的问题-一个人能找到一个精确的条件，保证逆散射问题是唯一可解的和运营商$I+H_{s_\pm}$是可逆的？一个人是否可以声称唯一性意味着可逆性，或者相反？此外，我们感兴趣的不仅是势的情况下，而且在渐近概周期势的情况下。因此，本文主要介绍Sturm-Liouville算子反问题的两种发展情况：（概）周期位势反问题和快速下降位势反问题。

The inverse problem for a spectral asymmetry function of the Schrödinger operator on a finite interval

For the Schrödinger equation $-d^2 u/dx^2 + q(x)u = λu$ on a finite $x$-interval, there is defined an "asymmetry function" $a(λ;q)$, which is entire of order $1/2$ and type $1$ in $λ$. Our main result identifies the classes of square-integrable potentials $q(x)$ that possess a common asymmetry function. For any given $a(λ)$, there is one potential for each Dirichlet spectral sequence.

ZS算子的反问题及其同构性

Inverse problems for ZS-operators and their isomorphisms

本文研究了ZS算子在单位区间和环上的同构反问题，通过非线性分析和4谱映射的性质，探讨了算子的同构性及其在不同边界条件下的应用。

On Inverse Spectral Problems for Second Order Integro-differential Operators

Inverse spectral problems are studied for the second order integro-differential operators on a finite interval. Properties of spectral characteristic are established, and the uniqueness theorem is proved for this class of inverse problems.

Inverse problems of recovering first-order integro-differential operators from spectra

Inverse spectral problems are studied for first-order integro-differential operators on a finite interval. These problems consist in recovering some components of the kernel from one or multiple spectra. Uniqueness theorems are proved for this class of inverse problems.

Reconstruction and Solvability for Discontinuous Hochstadt-Lieberman Problems

We consider Sturm-Liouville problems with a discontinuity in an interior point, which are motivated by the inverse problems for the torsional modes of the Earth. We assume that the potential on the right half-interval and the coefficient in the right boundary condition are given. Half-inverse problems are studied, that consist in recovering the potential on the left half-interval and the left boundary condition from the eigenvalues. If the discontinuity belongs to the left half-interval, the position and the parameters of the discontinuity also can be reconstructed. In this paper, we provide r…

有限区间上Dirac算子的谱逆问题

Inverse spectral problems for Dirac operators on a finite interval

本文考虑由微分表达式$$ \mathfrak t_q:=\frac{1}{i}[I&0 0&-I]\frac{d}{dx}+[0&q q^*&0] $$和一些分离的边界条件生成的Dirac算子的正谱问题和逆谱问题。这里，$q$是一个$r\times r$矩阵值函数，其条目属于$L_2((0,1),\mathbb C)$，$I$是单位$r\times r$矩阵。我们给出了一个完整的描述的光谱数据（特征值和适当引入的赋范矩阵）的运营商正在考虑和建议的算法重建潜在的$q$从相应的光谱数据。

Recovering Differential Operators with Nonseparated Boundary Conditions in the Central Symmetric Case

Inverse spectral problems for Sturm-Liouville operators on a finite interval with non-separated boundary conditions are studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss statements of the problems, provide algorithms for their solutions along with necessary and sufficient conditions for the solvability of the inverse problems considered.

通过柯西问题求解含Dirac权的逆Sturm-Liouville问题

Solving Inverse Dirac-weighted Sturm-Liouville Problems via Cauchy problems

本文基于点相互作用方法，通过构造柯西问题显式重构带单点、两点及多点Dirac权的正则Sturm-Liouville问题的势函数，并讨论多点情形。

Local solvability and stability of the inverse problem for the non-self-adjoint Sturm-Liouville operator

We consider the non-self-adjoint Sturm-Liouville operator on a finite interval. The inverse spectral problem is studied, which consists in recovering this operator from its eigenvalues and generalized weight numbers. We prove local solvability and stability of this inverse problem, relying on the method of spectral mappings. Possible splitting of multiple eigenvalues is taken into account.

有限Krein-Stieltjes弦的正问题与反问题：用点质量近似常数密度

Forward and inverse problems for a finite Krein-Stieltjes string. Approximation of constant density by point masses

本文研究Krein弦动态反问题，通过积分方程和Krein型方程恢复由有限点质量决定的弦密度，并探讨点质量近似常数密度及有限波速的影响。

内点具有Bessel型奇点的射线上Sturm-Liouville算子的反问题

An Inverse Problem for Sturm-Liouville Operators on the Half-line Having Bessel-type Singularity in an Interior Point

研究了从给定的Weyl函数恢复半直线上的Sturm-Liouville算子的反问题，该半直线上的Bessel型奇异性在区间内。证明了相应的唯一性定理，给出了反问题解的构造性步骤，得到了反问题可解的充要条件.