A note on one inverse spectral problem

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。

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

This is a correction to our article in the Journal of Differential Equations, Volume 76(1988).

Inverse spectral problem for the third-order differential equation

This paper is concerned with the inverse spectral problem for the third-order differential equation with distribution coefficient. The inverse problem consists in the recovery of the differential expression coefficients from the spectral data of two boundary value problems with separated boundary conditions. For this inverse problem, we solve the most fundamental question of the inverse spectral theory about the necessary and sufficient conditions of solvability. In addition, we prove the local solvability and stability of the inverse problem. Furthermore, we obtain very simple sufficient cond…

Direct and inverse spectral problems for rank-one perturbations of self-adjoint operators

For a given self-adjoint operator $A$ with discrete spectrum, we completely characterize possible eigenvalues of its rank-one perturbations~$B$ and discuss the inverse problem of reconstructing $B$ from its spectrum.

Inverse spectral problems for Sturm-Liouville operators with singular potentials, II. Reconstruction by two spectra

We solve the inverse spectral problem of recovering the singular potentials $q\in W^{-1}_{2}(0,1)$ of Sturm-Liouville operators by two spectra. The reconstruction algorithm is presented and necessary and sufficient conditions on two sequences to be spectral data for Sturm-Liouville operators under consideration are given.

A note on Spectral Analysis of Quantum graphs

We provide an introductory review of some topics in spectral theory of Laplacians on metric graphs. We focus on three different aspects: the trace formula, the self-adjointness problem and connections between Laplacians on metric graphs and discrete graph Laplacians.

正则哈密顿系统反谱问题中的周期逼近

Periodic approximations in inverse spectral problems for canonical Hamiltonian systems

本文研究半直线上正则哈密顿系统的反谱问题，通过谱测度周期化及对应哈密顿量的收敛性，将基于截断Toeplitz算子的周期算法推广至非周期测度情形。

Inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite tree

We consider inverse dynamic and spectral problems for the one dimensional Dirac system on a finite tree. Our aim will be to recover the topology of a tree (lengths and connectivity of edges) as well as matrix potentials on each edge. As inverse data we use the Weyl-Titchmarsh matrix function or the dynamic response operator.

A note on the geometric simplicity of the spectral radius of nonnegative irreducible tensors

We prove that the spectral radius of even order nonnegative irreducible tensors is real geometrically simple. In the case when the order of the tensor is odd, or in the complex field, some conditions are given to guarantee the geometric simplicity of the spectral radius.

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…

Survey of the inverse spectral problem for the Notices of the ICCM

This is an updated survey on the inverse spectral problem written for the Notices of the ICCM. It rapidly reviews some of the material in the previous survey of the same title (arXiv:math/0402356) and then discusses some relatively new results (rigidity results for the ellipse, phase shifts in scattering theory). The last section poses a number of open problems involving "inverse results for oscillatory integrals".

Jacobi矩阵的Ambarzumian型混合逆谱问题

Ambarzumian-type mixed inverse spectral problems for Jacobi matrices

本文研究Jacobi矩阵的Ambarzumian型混合逆谱问题，探讨在已知除前m个对角元素外的所有元素及m个有序特征值时矩阵的唯一确定性。

An inverse spectral problem for a star graph of Krein strings

We solve an inverse spectral problem for a star graph of Krein strings, where the known spectral data comprises the spectrum associated with the whole graph, the spectra associated with the individual edges as well as so-called coupling matrices. In particular, we show that these spectral quantities uniquely determine the weight within the class of Borel measures on the graph, which give rise to trace class resolvents. Furthermore, we obtain a concise characterization of all possible spectral data for this class of weights.

Trace formulae and inverse spectral theory for Schrödinger operators

We extend the well-known trace formula for Hill's equation to general one-dimensional Schrödinger operators. The new function $ξ$, which we introduce, is used to study absolutely continuous spectrum and inverse problems.

Inverse spectral problems for functional-differential operators with involution

The main goal of this paper is to propose an approach to inverse spectral problems for functional-differential operators (FDO) with involution. For definiteness, we focus on the second-order FDO with involution-reflection. Our approach is based on the reduction of the problem to the matrix form and on the solution of the inverse problem for the matrix Sturm-Liouville operator by developing the method of spectral mappings. The obtained matrix Sturm-Liouville operator contains the weight, which causes qualitative difficulties in the study of the inverse problem. As a result, we show that the con…

Inverse spectral theory for perturbed torus

We consider an inverse problem for Laplacians on rotationally symmetric manifolds, which are finite for the transversal direction and periodic with respect to the axis of the manifold, i.e., Laplacians on tori. We construct an infinite dimensional analytic isomorphism between the space of profiles (the radius of the rotation) of the torus and the spectral data as well as the stability estimates: those for the spectral data in terms of the profile and conversely, for the profile in term of the spectral data.

Inverse spectral problems for Sturm-Liouville operators with singular potentials

The inverse spectral problem is solved for the class of Sturm-Liouville operators with singular real-valued potentials from the space $W^{-1}_2(0,1)$. The potential is recovered via the eigenvalues and the corresponding norming constants. The reconstruction algorithm is presented and its stability proved. Also, the set of all possible spectral data is explicitly described and the isospectral sets are characterized.

On inverse spectral problems for Sturm-Liouville differential operators on closed sets

We study Sturm-Liouville operators on closed sets of a special structure, which are sometimes referred as time scales and often appear in modelling various real processes. Depending on the set structure, such operators unify both differential and difference operators. We obtain properties of their spectral characteristics and study inverse problems with respect to them. We prove that the spectral characteristics uniquely determine the operator.

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.

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.

Inverse Spectral Problem for Schrödinger Operators

In this article we improve some of the inverse spectral results proved by Guillemin and Uribe in \cite{GU}. They proved that under some symmetry assumptions on the potential $V(x)$, the Taylor expansion of $V(x)$ near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schrödinger operator in $\mathbb R^n$. We prove some similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of $V(x)$. We establish our results by findin…

Inverse spectral problem for singular AKNS operator on [0,1]

We consider an inverse spectral problem for a class of singular AKNS operators $H\_a, a\in\N$ with an explicit singularity. We construct for each $a\in\N$, a standard map $λ^a\timesκ^a$ with spectral data $λ^a$ and some norming constant $κ^a$. For $a=0$, $λ^a\timesκ^a$ was known to be a local coordinate system on $\lr\times\lr$. Using adapted transformation operators, we extend this result to any non-negative integer $a$, give a description of isospectral sets and we obtain a Borg-Levinson type theorem.

高阶微分算子逆谱问题的局部可解性与稳定性

Local solvability and stability of an inverse spectral problem for higher-order differential operators

本文首次证明了具有分布系数的高阶（n > 3）微分算子逆谱问题的局部可解性与稳定性，通过构造性方法将非线性逆问题转化为Banach空间中的线性方程，并证明了谱数据微小扰动下主方程的唯一可解性及系数差异估计。

An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator

The inverse problem of spectral analysis for the non-self-adjoint matrix Sturm-Liouville operator on a finite interval is investigated. We study properties of the spectral characteristics for the considered operator, and provide necessary and sufficient conditions for the solvability of the inverse problem. Our approach is based on the constructive solution of the inverse problem by the method of spectral mappings. The characterization of the spectral data in the self-adjoint case is derived as a corollary of the main result.

The inverse spectral problem for indefinite strings

Motivated by the study of certain nonlinear wave equations (in particular, the Camassa-Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form \[-u"=z\,u\,ω+z^2u\,\upsilon\] on an interval $[0,L)$, where $ω$ is a real-valued distribution in $H^{-1}_{\mathrm{loc}}[0,L)$, $\upsilon$ is a non-negative Borel measure on $[0,L)$ and $z$ is a complex spectral parameter. Apart from developing basic spectral theory for these kinds of spectral problems, our main result is an indefinite analogue of M. G. Krein's celebrated solution of …

关于作用于 $C^*$-代数的逆算子正性的一点注记

A note on the positivity of inverse operators acting on $C^*$-algebras

本文研究 $C^*$-代数上正可逆线性算子的逆算子正性问题，给出了 $T^{-1}$ 为正的充要条件，并通过反例说明即使 $T$ 为单位算子且谱在单位圆内，其逆算子也未必为正。

Inverse spectral theory for a class of non-compact Hankel operators

We characterize all bounded Hankel operators $Γ$ such that $Γ^*Γ$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data.

Survey of the inverse spectral problem

This is a survey of the inverse spectral problem on (mainly compact) Riemannian manifolds, with or without boundary. The emphasis is on wave invariants: on how wave invariants have been calculated and how they have been applied to concrete inverse spectral problems.

Rational interpolation and mixed inverse spectral problem for finite CMV matrices

For finite dimensional CMV matrices the mixed inverse spectral problem of reconstruction the matrix by its submatrix and a part of its spectrum is considered. A general rational interpolation problem which arises in solving the mixed inverse spectral problem is studied, and the description of the space of its solutions is given. We apply the developed technique to give sufficient conditions for the uniqueness of the solution of the mixed inverse spectral problem.

Direct and inverse spectral theory of singular left-definite Sturm-Liouville operators

We discuss direct and inverse spectral theory of self-adjoint Sturm-Liouville relations with separated boundary conditions in the left-definite setting. In particular, we develop singular Weyl-Titchmarsh theory for these relations. Consequently, we apply de Branges' subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa-Holm equation.

Half-inverse spectral problems for Sturm--Liouville operators with singular potentials

Half-inverse spectral problem for a Sturm--Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. We give the necessary and sufficient conditions for solvability of the half-inverse spectral problem for the class of Sturm--Liouville operators with singular potentials from the space $W^{-1}_2(0,1)$ and provide the reconstruction algorithm.

The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation

We solve the inverse spectral problem associated with periodic conservative multi-peakon solutions of the Camassa-Holm equation. The corresponding isospectral sets can be identified with finite dimensional tori.

An inverse spectral theory for finite CMV matrices

For finite dimensional CMV matrices the classical inverse spectral problems are considered. We solve the inverse problem of reconstructing a CMV matrix by its Weyl's function, the problem of reconstructing the matrix by two spectra of CMV matrices with different "boundary conditions", and the problem of reconstructing the CMV matrix by its spectrum and the spectrum of the CMV matrix obtained from it by truncation.

Inverse spectral problems for Dirac operators with summable matrix-valued potentials

We consider the direct and inverse spectral problems for Dirac operators on $(0,1)$ with matrix-valued potentials whose entries belong to $L_p(0,1)$, $p\in[1,\infty)$. We give a complete description of the spectral data (eigenvalues and suitably introduced norming matrices) for the operators under consideration and suggest a method for reconstructing the potential from the corresponding spectral data.

Inverse Indefinite Spectral Problem for Second Order Differential Operator with Complex Periodoc Coefficients

The inverse problem for the Sturm- Liouville operator with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is studied. We give formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.

Inverse spectral theory as influenced by Barry Simon

We survey Barry Simon's principal contributions to the field of inverse spectral theory in connection with one-dimensional Schrodinger and Jacobi operators.

一类非自伴带状矩阵的正谱问题与逆谱问题

Direct and inverse spectral problems for a class of non-selfadjoint band matrices

本文研究一类非自伴带状矩阵的正谱与逆谱问题，提出了从给定谱数据重构矩阵的充要条件，并推广了周期Jacobi矩阵等相关结论。

Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions

Inverse problems of recovering the coefficients of Sturm-Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

关于边界控制方法在谱估计与反问题中的若干应用

On some applications of the Boundary Control method to spectral estimation and inverse problems

本文将边界控制方法应用于广义谱估计与逆源问题，推导了相应方程，并揭示了其可解性依赖于动力系统可控性及指数族性质。

On a new inverse spectral problem

The differential operators pencil recovery uniqueness theorem is proved in our article. Novelty of this result is not equation coefficient recovery, but boundary conditions coefficients recovery. We show that all conditions of the theorem are essential. We also give some interesting examples.

逆谱理论的边界控制方法

The boundary control approach to inverse spectral theory

本文建立了逆谱问题多种方法间的联系，证明Boundary Control方法能以简单且物理动机明确的方式证明其他理论的核心结果，并推导了Gelfand--Levitan方程的局部版本。

An inverse spectral problem related to the Geng-Xue two-component peakon equation

We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral problems for those equations, this one is of a 'discrete cubic string' type -- a nonselfadjoint generalization of a classical inhomogeneous string -- but presents some interesting novel features: there are two Lax pairs, both of which contribute to the correct complete spectral data, and the solution to the inverse problem can be expressed using quantities relate…

Inverse spectral theory and the Minkowski problem for the surface of revolution

We solve the inverse spectral problem for rotationally symmetric manifolds, which include the class of surfaces of revolution, by giving an analytic isomorphism from the space of spectral data onto the space of functions describing the radius of rotation. An analogue of the Minkowski problem is also solved.

Inverse spectral problems for Sturm--Liouville operators with matrix-valued potentials

We give a complete description of the set of spectral data (eigenvalues and specially introduced norming constants) for Sturm--Liouville operators on the interval $[0,1]$ with matrix-valued potentials in the Sobolev space $W_2^{-1}$ and suggest an algorithm reconstructing the potential from the spectral data that is based on Krein's accelerant method.

Inverse Spectral Problems for Tridiagonal N by N Complex Hamiltonians

In this paper, the concept of generalized spectral function is introduced for finite-order tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix. The inverse problems from generalized spectral function as well as from spectral data are investigated. In this way, a procedure for construction of complex tridiagonal matrices having real eigenvalues is obtained.

An Inverse Spectral Theorem

We prove a substantial extension of an inverse spectral theorem of Ambarzumyan, and show that it can be applied to arbitrary compact Riemannian manifolds, compact quantum graphs and finite combinatorial graphs, subject to the imposition of Neumann (or Kirchhoff) boundary conditions.

Inverse problem of the spectral analysis for the Sturm-Liouville operator with non-separated boundary conditions and spectral parameter in the boundary condition

This work deals with an inverse problem for the Sturm-Liouville operator with non-separated boundary conditions, one of which linearly depends on a spectral parameter. Uniqueness theorem is proved, solution algorithm is constructed and sufficient conditions for solvability of inverse problem are obtained. As spectral data, we only use the spectrum of one boundary value problem and some sequence of signs.

Inverse spectral problems for Dirac operators with summable potentials

The spectral properties of Dirac operators on $(0,1)$ with potentials that belong entrywise to $L_p(0,1)$, for some $p\in[1,\infty)$, are studied. The algorithm of reconstruction of the potential from two spectra or from one spectrum and the corresponding norming constants is established, and a complete solution of the inverse spectral problem is provided.

Uniqueness results in the inverse spectral Steklov problem

This paper is devoted to an inverse Steklov problem for a particular class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. We prove that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance.

Analyticity and uniform stability in the inverse spectral problem for Dirac operators

We prove that the inverse spectral mapping reconstructing the square integrable potentials on [0,1] of Dirac operators in the AKNS form from their spectral data (two spectra or one spectrum and the corresponding norming constants) is analytic and uniformly stable in a certain sense.

Inverse problems in spectral geometry

In this survey we review positive inverse spectral and inverse resonant results for the following kinds of problems: Laplacians on bounded domains, Laplace-Beltrami operators on compact manifolds, Schrödinger operators, Laplacians on exterior domains, and Laplacians on manifolds which are hyperbolic near infinity.

具有双重广义Regge边界条件的薛定谔算子的正逆谱问题

Direct and inverse spectral problems for the Schrodinger operator with double generalized Regge boundary conditions

本文研究具有两个广义Regge边界条件的薛定谔算子的正逆谱问题，给出正问题谱的性质（含特征值渐近分布），证明逆问题中偶势、双谱及一般部分逆问题的唯一性定理。

On the factorization formula for fundamental solutions in the inverse spectral transform

A factorization formula for wave functions, which is basic in the inverse spectral transform approach to initial-boundary value problems, is proved in greater generality than before. Applications follow. Related compatibility questions for the GBDT version of Bäcklund-Darboux transformation are treated too.

Inverse spectral problems for energy-dependent Sturm-Liouville equations

We study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from their Dirichlet spectra and sequences of the norming constants. For the class of problems under consideration, we give a complete description of the corresponding spectral data, suggest a reconstruction algorithm, and establish uniqueness of reconstruction. The approach is based on connection between spectral problems for energy-dependent Sturm-Liouville equations and for Dirac operators of special form.