Inverse spectral problems in rectangular domains

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.

On a class of inverse electrostatic and elasticity problems

We study the inverse electrostatic and elasticity problems associated with Poisson and Navier equations. The uniqueness of solutions of these problems is proved for piecewise constant electric charge and internal stress distributions having a checkered structure: they are constant on rectangular blocks. Such distributions appear naturally in practical applications. We also discuss computational challenges arising in the numerical implementation of our method.

具有Hilfer导数的两项时间分数阶扩散方程的正问题和反源问题

Direct and inverse source problems for two-term time-fractional diffusion equation with Hilfer derivative

本文研究矩形域中具有两项广义分数阶导数（Hilfer导数）的扩散方程的正问题和反源问题，通过谱展开方法将问题转化为两项分数阶微分方程，并利用正弦函数和二元多项Mittag-Leffler函数表示解，在给定数据条件下证明无穷级数的一致收敛性。

Inverse source problems for degenerate time-fractional PDE

In this paper, we investigate two inverse source problems for degenerate time-fractional partial differential equation in rectangular domains. The first problem involves a space-degenerate partial differential equation and the second one involves a time-degenerate partial differential equation. Solutions to both problem are expressed in series expansions. For the first problem, we obtained solutions in the form of Fourier-Legendre series. Convergence and uniqueness of solutions have been discussed. Solutions to the second problem are expressed in the form of Fourier-Sine series and they involv…

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.

Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets

We consider the null-controllability problem for the generalized Baouendi-Grushin equation $(\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = 1_ωu$ on a rectangular domain. Sharp controllability results already exist when the control domain $ω$ is a vertical strip, or when $q(x) = x$. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $ω$. In some geometries for $ω$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability. O…

An inverse spectral problem for a fractional Schrödinger operator

We establish that the potential appearing in a fractional Schrödinger operator is uniquely determined by an internal spectral data.

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.

Some spectral properties and isoperimetric inequalities for a nonlocal Laplacian problem

In this paper we consider a non-local problem for a Laplace operator in a multidimensional bounded symmetric domain. The investigated problem is an analogue of the classical periodic boundary value problems in the case of non-rectangular domain. We prove self-adjointness of the problem and show a method of constructing eigenfunctions. We obtain an analogue of the Rayleigh type inequality and some spectral inequalities for the first eigenvalue of the nonlocal problem.

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.

On nonlinear boundary value problem corresponding to $N$-dimensional inverse spectral problem

We establish a relationship between an inverse optimization spectral problem for N-dimensional Schrödinger equation $ -Δψ+qψ=λψ$ and a solution of the nonlinear boundary value problem $-Δu+q_0 u=λu- u^{γ-1},~~u>0,~~ u|_{\partial Ω}=0$. Using this relationship, we find an exact solution for the inverse optimization spectral problem, investigate its stability and obtain new results on the existence and uniqueness of the solution for the nonlinear boundary value problem.

Semiclassical inverse spectral problem for elastic Love waves in isotropic media

We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we focus on Love waves. Under certain generic conditions, we establish uniqueness and present a reconstruction scheme for the S- wavespeed with multiple wells from the semiclassical spectrum of these waves.

Some Inverse Spectral Results in Exterior Transmission Problem

We consider an inverse spectral theory in a domain with the cavity that is bounded by a penetrable inhomogeneous medium. An ODE system is constructed piecewise through the solutions inside and outside the cavity. The ODE system is connected to the PDE system via the analytic continuation. For each scattered angle, we describe its eigenvalue density in the complex plane, and prove an inverse uniqueness on the inhomogeneity by the measurements in the far-fields.

反问题研究：具有非均匀狄利克雷边界数据的谱分数阶拉普拉斯算子

Inverse problems for the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data

本文研究了具有非均匀狄利克雷边界数据的谱分数阶拉普拉斯算子，提出狄利克雷到诺伊曼映射并分析反问题，同时建立谱分数阶拉普拉斯算子的密度结果。

Shape identification in inverse medium scattering problems with a single far-field pattern

Consider time-harmonic acoustic scattering from a bounded penetrable obstacle $D\subset \mathbb R^N$ embedded in a homogeneous background medium. The index of refraction characterizing the material inside $D$ is supposed to be Hölder continuous near the corners. If $D\subset \mathbb R^2$ is a convex polygon, we prove that its shape and location can be uniquely determined by the far-field pattern incited by a single incident wave at a fixed frequency. In dimensions $N \geq 3$, the uniqueness applies to penetrable scatterers of rectangular type with additional assumptions on the smoothness of th…

The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data

We establish the inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data. In particular, it is employed to prove existence of weak solutions for the corresponding Cauchy problem.

Construction of the solution of the inverse spectral problem for a system depending rationally on the spectral parameter, Borg-Marchenko-type theorem, and sine-Gordon equation

Weyl theory for a non-classical system depending rationally on the spectral parameter is treated. Borg-Marchenko-type uniqueness theorem is proved. The solution of the inverse problem is constructed. An application to sine-Gordon equation in laboratory coordinates is given.

Approximation of the inverse scattering Steklov eigenvalues and the inverse spectral problem

In this paper, we consider the numerical approximation of the Steklov eigenvalue problem that arises in inverse acoustic scattering. The underlying scattering problem is for an inhomogeneous isotropic medium. These eigenvalues have been proposed to be used as a target signature since they can be recovered from the scattering data. A Galerkin method is studied where the basis functions are the Neumann eigenfunctions of the Laplacian. Error estimates for the eigenvalues and eigenfunctions are proven by appealing to Weyl's Law. We will test this method against separation of variables in order to …