Steve Zelditch
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.
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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".
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.
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.
Global Solution of the Inverse Spectral Problem for Differential Operators on a Finite Interval with Complex Weights
本文研究有限区间上带复权因子的非自伴二阶微分算子的逆谱问题，建立了谱特征性质，提出基于谱映射方法的全局解构造算法。
The note contains the proof of the uniqueness theorem for the inverse problem in the case of $n$-th order differential equation.
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.
We solve the inverse problem from the spectral measure and the inverse three-spectra problem for the class of singular Krein strings on a finite interval with trace class resolvents. In particular, this includes a complete description of all possible spectral measures and three (Dirichlet) spectra associated with this class of Krein strings. The solutions of these inverse problems are obtained by approximation with Stieltjes strings.
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。
This is a correction to our article in the Journal of Differential Equations, Volume 76(1988).
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…
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.
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.
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.
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…
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.
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。
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。
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.
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.
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.
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…
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.
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.
We survey Barry Simon's principal contributions to the field of inverse spectral theory in connection with one-dimensional Schrodinger and Jacobi operators.
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.
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.
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.
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.
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…
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.
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.
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。
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.
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 …
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.
We consider a second order functional-differential pencil with two constant delays of the argument and study the inverse problem of recovering its coefficients from the spectra of two boundary value problems with one common boundary condition. The uniqueness theorem is proved and a constructive procedure for solving this inverse problem along with necessary and sufficient conditions for its solvability is obtained. Moreover, we give a survey on the contemporary state of the inverse spectral theory for operators with delay. The pencil under consideration generalizes Sturm-Liouville-type operato…
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.
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.
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.
This paper is related to an inverse problem for a class of Dirac operators with discontinuous coefficient and eigenvalue parameter contained in boundary conditions. The asymptotic formula of eigenvalues of this problem is examined. The theorem on completeness of eigenfunctions is proved. The expansion formula with respect to eigenfunctions is obtained and Parseval equality is given. Weyl solution and Weyl function are constructed. Uniqueness theorem of the inverse problem respect to the Weyl function is proved.
Regularization and inverse spectral problems for differential operators with distribution coefficients
本文研究具有分布系数的高阶微分算子，证明一类含 Mirzoev-Shkalikov 正则化矩阵的矩阵函数均与某微分表达式相关联，并构造关联矩阵族，进而将其用于从 Weyl-Yurko 矩阵和离散谱数据恢复分布系数的新型逆谱问题，证明唯一性定理并详细讨论 n=2 情形。
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.
Direct and inverse spectral continuity for Dirac operators
本文研究了半直线 Dirac 算子谱数据的同胚连续性，通过基于 Schur 算法求解半格上具 $δ$-相互作用算子的逆谱问题，首次证明了一般 $L^2$-情形下的显式双边一致估计。
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 Sturm-Liouville problem with singular potential and spectral parameter in the boundary conditions
本文首次研究边界条件含奇异势及谱参数多项式依赖的逆 Sturm-Liouville 问题，证明局部可解性与稳定性，给出子谱唯一性充要条件并建立重构方法。
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.
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。
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.
Uniform stability for the matrix inverse Sturm-Liouville problems
本文研究了矩阵斯图姆-刘维尔问题在有限区间和星型图上的均匀稳定性，通过构造性方法和谱映射技术，证明了反谱映射的有界性和相关估计。
Solving Barcilon's inverse problems by the method of spectral mappings
本文利用谱映射方法研究从三组谱恢复四阶微分算子的Barcilon反问题，建立了三组谱与Weyl-Yurko矩阵的关系，并在复值分布系数情形下证明了反问题解的唯一性定理。
Steklov Spectral Geometry for Annular Surfaces: Inverse spectral results and isospectral compactness
研究带边界紧致曲面上斯捷克洛夫谱的逆谱问题，通过分析谱泽塔函数等，证明截头圆锥侧面由斯捷克洛夫谱唯一确定，圆形环域也如此，且斯捷克洛夫等谱平坦环形曲面族在\(C^\infty\)拓扑中紧致。
The operator of double differentiation, perturbed by the composition of the differentiation operator and a convolution one, on a finite interval with Dirichlet boundary conditions is considered. We obtain uniform stability of recovering the convolution kernel from the spectrum in a weighted $L_2$-norm and in a weighted uniform norm. For this purpose, we successively prove uniform stability of each step of the algorithm for solving this inverse problem in both the norms. Besides justifying the numerical computations, the obtained results reveal some essential difference from the classical inver…
Partial Information for Inverse Spectral Uniqueness in Vibration System with Multiple Frozen Arguments
该研究针对含多个冻结参量的Sturm-Liouville算子逆谱问题，分析点集先验信息对振动系统谱唯一性的影响，验证无理独立性假设对逆谱唯一性的保障作用，为相关现场问题提供理论支撑。
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.
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.
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.
Partial Inverse Spectral Problems for Sturm-Liouville Operators with Frozen Arguments on a Star-Shaped Graph
本文研究星形图上具有冻结参数的 Sturm-Liouville 算子的部分逆谱问题，提出了一种基于向量函数系统 Riesz 基性质的构造性算法，利用部分谱数据重构单条边上的未知势函数。
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.
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.
Third-order inverse spectral problem with the three-point boundary conditions
研究从两组谱数据恢复三阶微分方程的唯一性和存在性，采用基于Weyl-Yurko矩阵的一般方法。
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