The "Action" Variable is not an Invariant for the Uniqueness in the Inverse Scattering Problem

Inverse scattering problem for Sturm-Liouville operator on non-compact A-graph. Uniqueness result

In a finite-dimensional Euclidian space we consider a connected metric graph with the following property: each two cycles can have at most one common point. Such graphs are called A-graphs. On noncompact A-graph we consider a scattering problem for Sturm-Liouville differential operator with standard matching conditions in the internal vertices. Transport, spectral and scattering problems for differential operators on graphs appear frequently in mathematics, natural sciences and engineering. In particular, direct and inverse problems for such operators are used to construct and study models in …

Inverse problems for Dirac operators with constant delay: uniqueness, characterization, uniform stability

We initiate studying inverse spectral problems for Dirac-type functional-differential operators with constant delay. For simplicity, we restrict ourselves to the case when the delay parameter is not less than one half of the interval. For the considered case, however, we give answers to the full range of questions usually raised in the inverse spectral theory. Specifically, reconstruction of two complex $L_2$-potentials is studied from either complete spectra or subspectra of two boundary value problems with one common boundary condition. We give conditions on the subspectra that are necessary…

An Inverse Uniqueness of a Phaseless Scattering Problem by Zero-Crossings

We discuss the inverse uniqueness problem in phaseless scattering by counting the zeros of its modulus of the scattering amplitude. The phase linearization of scattered wave field disturbs the originally uniform distribution of the zero set. There is a connection between the perturbation of the index of refraction and the zero distribution of the modulus. We conclude the inverse uniqueness of the phaseless problem from the point of view of interior transmission problem.

Uniqueness for Inverse Sturm-Liouville Problems with a Finite Number of Transmission Conditions

We establish various uniqueness results for inverse spectral problems of Sturm-Liouville operators with a finite number of discontinuities at interior points at which we impose the usual transmission conditions. We consider both the case of classical Robin and of eigenparameter dependent boundary conditions.

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.

Scattering theory for CMV matrices: uniqueness, Helson--Szegő and Strong SzegŐ theorems

We develop a scattering theory for CMV matrices, similar to the Faddeev--Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for the uniqueness, which are connected with the Helson--Szeg\H o and the Strong Szeg\H o theorems. The first condition is given in terms of the boundedness of a transformation operator associated to the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, correspo…

A Global Uniqueness on Spherically Stratified Dielectric Medium in Time-Harmonic Maxwell Equation with Interior Transmission Eigenvalues

A set of regularly distributed transmission eigenvalues generates a density function. We use such a density function inversely determines the form of the indicator function. Using the entire function theory, we reduce an uniqueness problem with interior transmission eigenvalues induced by time-harmonic Maxwell equation to an uniqueness problem in entire function theory. In such an inverse problem, the definite integral of the square root of refraction index is the main parameter.

星图上非局部 Sturm-Liouville 算子逆谱问题的唯一性

Uniqueness of Inverse Spectral Problem of Non-Local Sturm-Liouville Operators on Star Graph

本文研究了星形图上带非局部势的 Sturm-Liouville 算子逆谱问题，在顶点引入冻结参数条件模拟网络控制中心，证明了系统在非局部边界条件下的逆唯一性与可解性，并给出了 Weyl 型谱渐近性质。

Uniqueness on the Class of Odd-Dimensional Starlike Obstacles with Cross Section Data

We determine the uniqueness on starlike obstacles by using the cross section data. We see cross section data as spectral measure in polar coordinate at far field. Cross section scattering data suffice to give the local behavior of the wave trace. These local trace formulas contain the geometric information on the obstacle. Local wave trace behavior is connected to the cross section scattering data by Lax-Phillips' formula. Once the scattering data are identical from two different obstacles, the short time behavior of the localized wave trace is expected to give identical heat/wave invariants.

不相交数据反问题中的唯一性与非唯一性：一个尖锐的谱阈值

Uniqueness and nonuniqueness in the disjoint data inverse problem: a sharp spectral threshold

该研究探讨了从波观测确定流形的反问题，证明了在次临界谱下界条件下唯一性成立，并构造了临界端点处唯一性失效的反例。

Inverse uniqueness results for one-dimensional weighted Dirac operators

Given a one-dimensional weighted Dirac operator we can define a spectral measure by virtue of singular Weyl-Titchmarsh-Kodaira theory. Using the theory of de Branges spaces we show that the spectral measure uniquely determines the Dirac operator up to a gauge transformation. Our result applies in particular to radial Dirac operators and extends the classical results for Dirac operators with one regular endpoint. Moreover, our result also improves the currently known results for canonical (Hamiltonian) systems. If one endpoint is limit circle case, we also establish corresponding two-spectra re…