Erratum for: Inverse Spectral Theory for a Singular Sturm-Liouville Operator on [0,1]

Spectral properties of Sturm-Liouville equations with singular energy-dependent potentials

We study spectral properties of energy-dependent Sturm-Liouville equations, introduce the notion of norming constants and establish their interrelation with the spectra. One of the main tools is the linearization of the problem in a suitable Pontryagin space.

The regularized trace formula for the Sturm-Liouville equation with spectral parameter in the boundary conditions

The regularized trace formula of first order for the Sturm-Liouville equation with spectral parameter in the boundary conditions is obtained.

A priori bounds and existence of non-real eigenvalues of indefinite Sturm-Liouville problems

The present paper gives a priori bounds on the possible non-real eigenvalues of regular indefinite Sturm-Liouville problems and obtains sufficient conditions for such problems to admit non-real eigenvalues.

On Properties of the Sturm-Liouville Operator with Degenerate Boundary Conditions

We consider spectral problems for the Sturm-Liouville operator with arbitrary complex-valued potential q(x) and degenerate boundary conditions. We solve corresponding inverse problem, and also study the completeness property and the basis property of the root function system.

On an inverse problem for Sturm-Liouville equation

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

In this work, we use the \textit{regularized sampling method} to compute the eigenvalues of Sturm Liouville problems with discontinuity conditions inside a finite interval. We work out an example by computing a few eigenvalues and their corresponding eigenfunctions.

Discrete Fractional Sturm-Liouville Equations

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

The author tries to derive the asymptotic expression of the large eigevalues of some vectorial Sturm-Liouville differential equations. A precise description for the formula of the square root of the large eiegnvalues up to the $O(1/n)$-term is obtained.

Estimates on the non-real eigenvalues of regular indefinite Sturm-Liouville problems

Regular Sturm-Liouville problems with indefinite weight functions may possess finitely many non-real eigenvalues. In this note we prove explicit bounds on the real and imaginary parts of these eigenvalues in terms of the coefficients of the differential expression.

Constructive solution of the inverse spectral problem for the matrix Sturm-Liouville operator

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

In this paper we study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from two spectra. We give a reconstruction algorithm and establish existence and uniqueness of reconstruction. Our approach essentially exploits the connection between the spectral problems under study and those for Dirac operators of a special form.

An application of the fixed point theorem to the inverse Sturm-Liouville problem

We consider Sturm-Liouville operators $-y''+v(x)y$ on $[0,1]$ with Dirichlet boundary conditions $y(0)=y(1)=0$. For any $1\le p<\infty$, we give a short proof of the characterization theorem for the spectral data corresponding to $v\in L^p(0,1)$.

Inverse problems of spectral analysis for the Sturm-Liouville operator with regular boundary conditions

We consider the Sturm-Liouville operator Lu=u''-q(x)u with regular but not strongly regular boundary conditions. Under some supplementary assumptions we prove that the set of potentials q(x) that ensure an asymptotically multiple spectrum is everywhere dense in the space of summable functions.

On recovering the Sturm--Liouville differential operators on time scales

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

We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with general regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root functions of these operators do not form a Riesz basis.

Inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems

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…

Inverse Sturm-Liouville problem with analytical functions in the boundary condition

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

We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with some regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root functions of these operators do not form a Riesz basis.

Half Inverse Sturm-Liouville Problem on Three-Star Graph with a Discontinuity

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

We consider the nonselfadjoint Sturm-Liouville operator Lu=u''-q(x)u with regular but not strongly regular boundary conditions, where q(x)is an arbitrary complex-valued summable function. We examine the basis property for the root function system of the mentioned operator.

Spectral Analysis for the Matrix Sturm-Liouville Operator on a Finite Interval

The inverse spectral problem is investigated for the matrix Sturm-Liouville equation on a finite interval. Properties of spectral characteristics are provided, a constructive procedure for the solution of the inverse problem along with necessary and sufficient conditions for its solvability is obtained.

Direct and inverse problems for the matrix Sturm-Liouville operator with the general self-adjoint boundary conditions

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.

具有冻结变量的Sturm-Liouville算子恢复的一致稳定性

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。

Inverse Spectral Problems for Sturm-Liouville Operators on Hedgehog-type Graphs with General Matching Conditions

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.

An inverse spectral problem for the matrix Sturm-Liouville operator on the half-line

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.

具有多个冻结参数的Sturm-Liouville方程的反谱问题

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。

On some class of singular Sturm-Liouville problems

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

This paper deals with the computation of the eigenvalues of non self-adjoint Sturm-Liouville problems with parameter dependent boundary conditions using the \textit{regularized sampling method}. A few numerical examples among which singular ones will be presented to illustrate the merit of the method and comparison made with the exact eigenvalues when they are available.

On the Nonself-adjoint Sturm-Liouville Operator with Matrix Potential

In this article we obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the nonself-adjoint operator generated by a system of Sturm-Liouville equations with summable coefficients and the quasiperiodic boundary conditions. Then using these asymptotic formulas, we find the conditions on the potential for which the root funcions of this operator form a Riesz basis.

星形图上逆斯图尔姆-刘维尔问题的均匀稳定性

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。

Eigenvalues of Sturm-Liouville Operators with Distributional Potentials

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…

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.

Sturm-Liouville Difference Equations Having Special Potentials

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.

Spectral data asymptotics for the matrix Sturm-Liouville operator

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 …

A uniqueness theorem on inverse spectral problems for the Sturm--Liouville differential operators on time scales

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

In this work we investigate the resolvent operator and completeness of eigenfunctions of a Sturm-Liouville problem with discontinuities at two points. The problem contains an eigenparameter in the one of boundary conditions. For operator-theoretic formulation of the considered problem we define an equivalent inner product in the Hilbert space and suitable self-adjoint lineer operator in it.

Solving an inverse problem for the Sturm-Liouville operator with a singular potential by Yurko's method

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

Transfer functions and local spectral uniqueness for Sturm-Liouville operators, canonical systems and strings

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

In this paper we study a Sturm--Liouville operator $Ly=-y"+q(x)y$ in the space $L_2[0,π]$ with Direchlet boundary conditions. Here the potential $q$ is a first order distribution: $q\in W_2^{-1}[0,π]$. Such operators were defined in our previous papers. Here we clerify two leading terms in asymptotic formulae for eigenfunctions of such operators and for functions of biorthogonal system.

On Povzner--Wienholtz-type Self-Adjointness Results for Matrix-Valued Sturm--Liouville Operators

We derive Povzner--Wienholtz-type self-adjointness results for $m\times m$ matrix-valued Sturm--Liouville operators $T=R^{-1}\big[-\f{d}{dx}P\f{d}{dx}+Q\big]$ in $L^2((a,b);Rdx)^m$, $m\in\bbN$, for $(a,b)$ a half-line or $\bbR$.

Analytic continuation and resonance-free regions for Sturm-Liouville pote ntials with power decay

We are concerned with the Sturm-Liouville problem on the half line. We show that when the potential $q$ is subject only to power decay at infinity the $L^2$ solution may be continued into a sector of the so-called un-physical sheet. This gives rise to resonance free regions and the numerical estimation of resonances outside these regions.

Inverse problem solution and spectral data characterization for the matrix Sturm-Liouville operator with singular potential

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…

A partial inverse problem for the Sturm-Liouville operator on the graph with a loop

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.

Partial inverse problems for Sturm-Liouville operators on trees

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

This paper deals with the computation of the eigenvalues of two-parameter Sturm- Liouville (SL) problems using the Regularized Sampling Method, a method which has been effective in computing the eigenvalues of broad classes of SL problems (Singular, Non-Self-Adjoint, Non-Local, Impulsive,...). We have shown, in this work that it can tackle two-parameter SL problems with equal ease. An example was provided to illustrate the effectiveness of the method.

Asymptotic behaviour of the positive spectrum for a family of periodic Sturm-Liouville problems in the transition from a definite problem to indefinite one

Spectral problem for a family of periodic Sturm--Liouville problems \[ u''+λ^2(a(x)-a)u=0 \] depending on the parameter (a\in\mathbb R) is considered. An interpolation formula describing the behaviour of the branches of the spectrum in the transition from a definite problem to indefinite one is obtained.

关于Sturm-Liouville算子基本特征值间隙

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。

带脉冲的向量 Sturm-Liouville 算子的 Ambarzumyan 型定理

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

In this study, we give a regular fractional Sturm Liouville problem for diffusion operator (FSLPDO), research the spectral properties of the eigenfunctions and eigenvalues of the diffusion operator. We show that the eigenvalues and eigenfunctions of (FSLPDO) are real and orthogonal, respectively and fractional diffusion operator is self adjoint.

On two-point boundary value problems for the Sturm-Liouville operator

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

Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered.

准周期逆Sturm-Liouville问题

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

In this article we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operator generated in space of vector-functions by the Sturm-Liouville equation with m by m matrix potential and the boundary conditions whose scalar case (m=1) are strongly regular. Using these asymptotic formulas, we find a condition on the potential for which the root functions of this operator form a Riesz basis.

Inverse Scattering Problem for a Piecewise Continuous Sturm - Liouville Equation with Eigenparameter Dependence in the Boundary Conditon

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.