Natalia P. Bondarenko
The inverse spectral problems are studied for the Sturm-Liouville operator on the star-shaped graph and for the matrix Sturm-Liouville operator with the boundary condition in the general self-adjoint form. We obtain necessary and sufficient conditions of solvability for these two inverse problems, and also prove their local solvability and stability.
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On the Construction of Isospectral Vectorial Sturm-Liouville Differential Equations
本文将Jodeit和Levitan构造经典标量Sturm-Liouville微分方程等谱问题的思想推广到向量Sturm-Liouville微分方程.发现了一些有趣的关系。
The inverse scattering problem is studied for the matrix Sturm-Liouville equation on the line. Necessary and sufficient conditions for the scattering data are obtained.
The self-adjoint and $m$-sectorial extensions of coercive Sturm-Liouville operators are characterised, under minimal smoothness conditions on the coefficients of the differential expression.
We consider Sturm-Liouville operators with singular potentials from the class on star-type graph with cycle, which consist the edges with commensurable lengths. Asymptotic representation for eigenvalues for such operators is obtained. Recovering of the characteristic function the Sturm-Liouville operators with the singular potentials is considered.
Inverse spectral problems for Sturm-Liouville operators with singular potentials, II. Reconstruction by two spectra
解决了由两个谱恢复Sturm-Liouville算子奇异势$q\in W^{-1}_{2}(0,1)$的逆谱问题。给出了重建算法，并给出了Sturm-Liouville算子谱数据的充要条件。
Spectral properties of Sturm-Liouville equations with singular energy-dependent potentials
研究了能量依赖的Sturm-Liouville方程的谱性质，引入了赋范常数的概念，建立了赋范常数与谱的关系。其中一个主要的工具是线性化的问题在一个合适的庞特里亚金空间。
The regularized trace formula for the Sturm-Liouville equation with spectral parameter in the boundary conditions
得到了在边界条件下带谱参数的Sturm-Liouville方程的一阶正则迹公式。
A priori bounds and existence of non-real eigenvalues of indefinite Sturm-Liouville problems
本文给出了正则不定Sturm-Liouville问题的可能非实特征值的先验界，并得到了这类问题允许非实特征值存在的充分条件.
On Properties of the Sturm-Liouville Operator with Degenerate Boundary Conditions
本文研究了具有任意复值位势q（x）和退化边界条件的Sturm-Liouville算子的谱问题.我们求解了相应的反问题，并研究了根函数系的完备性和基的性质。
In this study, the theorem on necessary and sufficient conditions for the solvability of inverse problem for Sturm-Liouville operator with discontinuous coefficient is proved and the algorithm of reconstruction of potential from spectral data (eigenvalues and normalizing numbers) is given.
Eigenvalues of Sturm Liouville problems with discontinuity conditions inside a finite interval
在这项工作中，我们使用\textit{regularized sampling method}来计算Sturm-Liouville问题的特征值与间断条件在有限的时间间隔内。通过计算几个本征值及其对应的本征函数，给出了一个例子。
In this study, we define discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Grünwald-Letnikov fractional operators with both delta and nabla operators. We show selfadjointness of the DFSL operator for the first time and prove some spectral properties, like orthogonality of distinct eigenfunctions, reality of eigenvalues, paralelly in integer and fractional order differential operator counterparts.
On The Eigenvalues of Some Vectorial Sturm-Liouville Eigenvalue Problems
本文试图导出某些向量Sturm-Liouville微分方程的大特征值的渐近表达式。得到了直到$O(1/n)$项的大特征值平方根公式的精确描述。
Erratum for: Inverse Spectral Theory for a Singular Sturm-Liouville Operator on [0,1]
这是对我们在《微分方程杂志》第76卷（1988年）中的文章的更正。
Estimates on the non-real eigenvalues of regular indefinite Sturm-Liouville problems
具有不定权函数的正则Sturm-Liouville问题可能具有多个非实特征值。本文证明了这些特征值的实部和虚部的显式界。
An inverse spectral problem is studied for the matrix Sturm-Liouville operator on a finite interval with the general self-adjoint boundary condition. We obtain a constructive solution based on the method of spectral mappings for the considered inverse problem. The nonlinear inverse problem is reduced to a linear equation in a special Banach space of infinite matrix sequences. In addition, we apply our results to the Sturm-Liouville operator on a star-shaped graph.
Reconstruction of energy-dependent Sturm-Liouville equations from two spectra
本文研究了由两个谱重构能量相关的Sturm-Liouville方程的逆谱问题。我们给出了一个重建算法，并证明了重建的存在性和唯一性。我们的方法基本上利用了研究中的谱问题和特殊形式的狄拉克算子之间的联系。
An application of the fixed point theorem to the inverse Sturm-Liouville problem
本文考虑了$[0,1]$上的Sturm-Liouville算子$-y''+v(x)y$，其Dirichlet边界条件为$y(0)=y(1)=0$ .对于任何$1\le p<\infty$，我们给出了对应于$v\in L^p(0,1)$的谱数据的特征定理的一个简短证明.
Inverse problems of spectral analysis for the Sturm-Liouville operator with regular boundary conditions
考虑Sturm-Liouville算子Lu= u“-q（x）u具有正则但非强正则边界条件.在一些补充的假设下，我们证明了在可和函数空间中，保证渐近多重谱的势集q（x）处处稠密.
We study Sturm--Liouville differential operators on the time scales consisting of a finite number of isolated points and segments. In a previous paper it was established that such operators are uniquely determined by their spectral characteristics. In the present paper, an algorithm for their recovery based on the method of spectral mappings is obtained. We also prove that the eigenvalues of two Sturm--Liouville boundary value problems with one common boundary condition alternate.
On the Basis Property of the Root Functions of Sturm-Liouville Operators with General Regular Boundary Conditions
得到了Sturm-Liouville算子在一般正则边界条件下本征值和本征函数的渐近公式。利用这些公式，我们找到了关于势q的充分条件，使得这些算子的根函数不形成Riesz基。
In this paper, inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems are established. For a fixed discrete Sturm-Liouville equation, inequalities among eigenvalues for different boundary conditions are given. For a fixed boundary condition, inequalities among eigenvalues for different equations are given. These results are obtained by applying continuity and discontinuity of the n-th eigenvalue function, monotonicity in some direction of the n-th eigenvalue function, which were given in our previous papers, and natural loops in the space of boundary conditi…
The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness, and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.
On the Basis Property of the Root Functions of Some Class of Non-self-adjoint Sturm--Liouville Operators
得到了Sturm-Liouville算子在某些正则边界条件下本征值和本征函数的渐近公式。利用这些公式，我们找到了关于势q的充分条件，使得这些算子的根函数不形成Riesz基。
We consider the inverse Sturm-Liouville problem with one discontinuous point on three-star graph, we deduced the distribution of the eigenvalues, and proved that one spectrum could uniquely determine the unknown potential and jump information when we know the whole potentials on two edges and half potential on another edge.
On the basis property of the root function systems of regular boundary value problems for the Sturm-Liouville operator
考虑非自伴Sturm-Liouville算子Lu= u“-q（x）u具有正则但非强正则边界条件，其中q（x）是任意复值可和函数.我们研究了上述算子的根函数系的基性质。
Spectral Analysis for the Matrix Sturm-Liouville Operator on a Finite Interval
研究了有限区间上矩阵Sturm-Liouville方程的逆谱问题。给出了谱特征的性质，构造了反问题的求解方法，并给出了反问题可解的充要条件.
The matrix Sturm-Liouville operator on a finite interval with the boundary conditions in the general self-adjoint form and with the singular potential from the class $W_2^{-1}$ is studied. This operator generalizes Sturm-Liouville operators on geometrical graphs. We investigate structural and asymptotical properties of the spectral data (eigenvalues and weight matrices) of this operator. Furthermore, we prove the uniqueness of recovering the operator from its spectral data, by using the method of spectral mappings.
Uniform stability of recovering the Sturm-Liouville operators with frozen argument
本文研究从谱及附加数据反演具有冻结变量的Sturm-Liouville算子，证明势函数对输入数据具有Lipschitz连续依赖的一致稳定性。
Boundary value problems on hedgehog-type graphs for Sturm-Liouville differential operators with general matching conditions are studied. We investigate inverse spectral problems of recovering the coefficients of the differential equation from the spectral data. For this inverse problem we prove a uniqueness theorem and provide a procedure for constructing its solution.
The matrix Sturm-Liouville operator with an integrable potential on the half-line is considered. We study the inverse spectral problem, which consists in recovering of this operator by the Weyl matrix. The main result of the paper is the necessary and sufficient conditions on the Weyl matrix of the non-self-adjoint matrix Sturm-Liouville operator. We also investigate the self-adjoint case and obtain the characterization of the spectral data as a corollary of our general result.
Inverse spectral problem of Sturm-Liouville equation with many frozen arguments
本文研究具有多个冻结参数的Sturm-Liouville算子的反谱问题，在假设条件下证明唯一性定理，并给出数值模拟。
Sturm-Liouville spectral problem for equation $-(y'/r)'+qy=λpy$ with generalized functions $r\ge 0$, $q$ and $p$ is considered. It is shown that the problem may be reduced to analogous problem with $r\equiv 1$. The case of $q=0$ and self-similar $r$ and $p$ is considered as an example.
Computing the spectrum of non self-adjoint Sturm-Liouville problems with parameter dependent boundary conditions
本文讨论了用\textit{regularized sampling method}计算具有参数依赖边界条件的非自伴Sturm-Liouville问题的特征值。文中给出了几个数值算例，其中包括奇异值的计算，以说明本方法的优点，并与可用的精确特征值进行了比较。
On the Nonself-adjoint Sturm-Liouville Operator with Matrix Potential
本文给出了具有可和系数和拟周期边界条件的Sturm-Liouville方程组生成的非自伴随算子的本征值和本征函数的渐近公式。然后利用这些渐近公式，我们找到了该算子的根函数形成Riesz基的势的条件。
Uniform stability of the inverse Sturm-Liouville problem on a star-shaped graph
本文研究了星形图上逆斯图尔姆-刘维尔问题的均匀稳定性，通过谱数据恢复势函数并证明其稳定性。
We introduce a novel approach for dealing with eigenvalue problems of Sturm-Liouville operators generated by the differential expression \begin{equation*} Ly=\frac{1}{r}\left( -(p\left[ y^{\prime }+sy\right] )^{\prime }+sp\left[ y^{\prime }+sy\right] +qy\right) \end{equation*} which is based on norm resolvent convergence of classical Sturm-Liouville operators. This enables us to describe the continuous dependence of the $n$-th eigenvalue on the space of self-adjoint boundary conditions and the coefficients of the differential equation after giving the inequalities among the eigenvalues. Moreov…
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.
In this paper, we present a new approachment for Sturm-Liouville problem having special potentials. We acquire the representations of solutions and asymptotic formulas for solutions with regard to initial conditions. Also, a few applications are given to show the requirement of Sturm-Liouville difference equations having potential function in view of suitability to the spectral theory. The approximate numerical outcomes for the eigenfunctions are compared with each other.
The self-adjoint matrix Sturm-Liouville operator on a finite interval with a boundary condition in the general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm-Liouville operators on a star-shaped graph with two different types of matching …
In the paper, Sturm--Liouville differential operators on time scales consisting of a finite number of isolated points and segments are considered. Such operators unify differential and difference operators. We obtain properties of their spectral characteristics including asymptotic formulae for eigenvalues and weight numbers. Uniqueness theorem is proved for recovering the operators from the spectral characteristics.
Resolvent Operator and Eigenfunctions of a Sturm-Liouville Type problem
本文研究了两点间断的Sturm-Liouville问题的预解算子和特征函数的完备性。该问题的边界条件之一包含一个特征参数。对于所考虑的问题的算子理论公式，我们定义了一个等价的内积在希尔伯特空间和适当的自伴线性算子。
An inverse spectral problem for the Sturm-Liouville operator with a singular potential from the class $W_2^{-1}$ is solved by the method of spectral mappings. We prove the uniqueness theorem, develop a constructive algorithm for solution, and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and the non-self-adjoint cases
It is shown that transfer functions, which play a crucial role in M.G. Krein's study of inverse spectral problems, are a proper tool to formulate local spectral uniqueness conditions.
On the eigenfunctions of Sturm--Liouville operators with potentials --- distributions
本文研究了空间$L_2[0,π]$中具有Direchlet边界条件的Sturm-Liouville算子$Ly=-y"+q(x)y$.这里，潜在的$q$是一阶分布：$q\in W_2^{-1}[0,π]$。这些算子在我们以前的论文中已经定义过了。本文阐明了这类算子的本征函数和双正交系函数的渐近公式中的两个首项。
On Povzner--Wienholtz-type Self-Adjointness Results for Matrix-Valued Sturm--Liouville Operators
本文给出了$m\times m$矩阵值Sturm-Liouville算子$T=R^{-1}\big[-\f{d}{dx}P\f{d}{dx}+Q\big]$在$L^2((a,b);Rdx)^m$，$m\in\bbN$，$(a,b)$为半直线或$\bbR$中的Povzner-Wienholtz型自伴性结果.
Analytic continuation and resonance-free regions for Sturm-Liouville pote ntials with power decay
我们关心的是射线上的Sturm-Liouville问题。我们表明，当潜在的$q$只受到功率衰减在无穷大的$L^2$解决方案可能会继续到一个部门的所谓的非物理片。这就产生了无共振区域和这些区域之外的共振的数值估计。
The matrix Sturm-Liouville operator on a finite interval with singular potential of class $W_2^{-1}$ and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm-Liouville operators on geometrical graphs. We investigate the inverse problem that consists in recovering the considered operator from the spectral data (eigenvalues and weight matrices). The inverse problem is reduced to a linear equation in a suitable Banach space, and a constructive algorithm for the inverse problem solution is developed. Moreover, we obtain the spectral data characterization for…
The Sturm-Liouville operator with singular potentials on the lasso graph is considered. We suppose that the potential is known a priori on the boundary edge, and recover the potential on the loop from a part of the spectrum and some additional data. We prove the uniqueness theorem and provide a constructive algorithm for the solution of this partial inverse problem.
In this paper, inverse spectral problems for Sturm-Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b - 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.
Two-parameter Sturm-Liouville problems
本文讨论了用正则化抽样方法计算两参数Sturm-Liouville（SL）问题的特征值，该方法在计算SL问题（奇异、非自伴、非局部、脉冲等）的特征值方面是有效的。我们已经表明，在这项工作中，它可以解决两个参数SL问题同样容易。最后通过算例说明了该方法的有效性。
Asymptotic behaviour of the positive spectrum for a family of periodic Sturm-Liouville problems in the transition from a definite problem to indefinite one
考虑一族周期Sturm-Liouville问题的谱问题\[ u''+λ^2(a(x)-a)u=0 \]取决于参数（\in \mathbb R）。得到了描述谱分支在从定问题到不定问题过渡过程中的行为的插值公式。
On the Fundamental Eigenvalues gap of Sturm-Liouville Operators
本文利用直接优化方法，在单阱势和单垒权函数及凸性约束下，研究区间上Sturm-Liouville算子基本特征值间隙的极小化子。
Ambarzumyan-type theorem for vectorial Sturm-Liouville operator with impulses
本文研究了带 Neumann 条件的向量脉冲 Sturm-Liouville 问题，证明了其 Ambarzumyan 定理，即若问题特征值与零势一致则势为零。
Spectral Properties of Fractional Sturm-Liouville Problem for Diffusion Operator
本文给出了扩散算子的一个正则分数阶SturmLiouville问题（FSLPDO），研究了该扩散算子的本征函数和本征值的谱性质.证明了（FSLPDO）的特征值和特征函数分别是实的和正交的，并且分数阶扩散算子是自伴的.
In this paper, we study spectral problems for the Sturm-Liouville operator with arbitrary complexvalued potential q(x) and two-point boundary conditions. All types of mentioned boundary conditions are considered. We ivestigate in detail the completeness property and the basis property of the root function system.
A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators
在权函数具有一个转向点的假设下，研究了加权Sturm-Liouville算子的本质谱的特征值.一个抽象的方法来解决这个问题，通过一个功能模型的不定Sturm-Liouville算子。得到了特征值的代数重数。此外，考虑了具有有限个奇异临界点的算子。
Quasiperiodic inverse Sturm-Liouville problem
本文针对具有非分离准周期边界条件的Sturm-Liouville逆谱问题，从Hill型判别式等信息恢复问题参数，得到其可解性及稳定性相关结论。
On the Non-Self-adjoint Sturm-Liouville Operators in the Space of Vector-Functions
本文给出了具有m × m矩阵势的Sturm-Liouville方程在向量函数空间中生成的非自伴算子的本征值和本征函数的渐近公式，其标量情形（m=1）是强正则的.利用这些渐近公式，我们找到了一个条件的潜力，该运营商的根函数形成一个Riesz基。
In this study, the inverse problem of the scattering theory on the half line for a piecewise continuous Sturm-Liouville equation with boundary condition depending quadratic on the spectral parameter is considered. The scattering data of the problem is defined, some properties of the scattering data are investigated. The main equation is derived and uniqueness algorithm to the potential with given scattering data is studied.
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