相关论文: Property C for ODE and applications

We consider the problem of recovering a smooth, compactly supported potential on R^3 from its backscattering data. We show that if two such potentials have the same backscattering data and the difference of the two potentials has controlled…

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…

In recent years, there appeared a considerable interest in the inverse spectral theory for functional-differential operators with constant delay. In particular, it is well known that, for each fixed $\nu\in\{0,1\},$ the spectra of two…

An inverse problem of finding an obstacle and the boundary condition on its surface from the fixed-energy scattering data is studied. A new method is developed for a proof of the uniqueness results. The method does not use the discreteness…

An approach for solving a variety of inverse coefficient problems for the Sturm-Liouville equation -y''+q(x)y={\lambda}y with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations…

Given a set of transmission eigenvalues, we describe the connection of such a set to the indicator functions in entire function theory. The indicator functions control the asymptotic growth rate of the solution of the Sturm-Liouville…

We propose a discrete approach for solving an inverse problem for Schr\"odinger's equation in two dimensions, where the unknown potential is to be determined from boundary measurements of the Dirichlet to Neumann map. For absorptive…

We consider two main inverse Sturm-Liouville problems: the problem of recovery of the potential and the boundary conditions from two spectra or from a spectral density function. A simple method for practical solution of such problems is…

We give an overview of recent developments in Sturm-Liouville theory concerning operators of transmutation (transformation) and spectral parameter power series (SPPS). The possibility to write down the dispersion (characteristic) equations…

The inverse scattering problem of the three-dimensional Schroedinger equation is considered at fixed scattering energy with spherically symmetric potentials. The phase shifts determine the potential therefore a constructive scheme for…

In this work, we consider the Sturm-Liouville operator on a finite interval $[0,1]$ with discontinuous conditions at $1/2$. We prove that if the potential is known a priori on a subinterval $[b,1]$ with $b\ge1/2$, then parts of two spectra…

The paper is denoted to the initial-boundary value problem for the wave equation with the Sturm-Liouville operator with irregular (distributive) potentials. To obtain a solution to the equation, the separation method and asymptotics of the…

Motivated by inverse problems with a single passive measurement, we introduce and analyze a new class of inverse spectral problems on closed Riemannian manifolds. Specifically, we establish two general uniqueness results for the recovery of…

We analyze the spectral properties of a self-adjoint second-order differential operator $\hat{C}$, defined on the Hilbert space $L^2([-v_c, v_c])$ with Dirichlet boundary conditions. We derive the discrete spectrum $\{C_n\}$, prove the…

A review of some of the author's results in the area of inverse scattering is given. The following topics are discussed: 1) Property $C$ and applications, 2) Stable inversion of fixed-energy 3D scattering data and its error estimate, 3)…

In this paper, we explore the inverse spectral problem of Sturm-Liouville operator on a star-like graph. To this fixed star-like graph centered at the origin as its vertex, we attach $m$ edges. On each edge, we impose the Sturm-Liouville…

Let $\ell_j:=-\frac{d^2}{dx^2}+k^2q_j(x),$ $k=const>0, j=1,2,$ $0<c_0\leq q_j(x)\leq c_1,$ %$q\in BV([0,1])$, $q$ has finitely many discontinuity points $x_m\in [0,1],$ and is real-analytic on the intervals $[x_m,x_{m+1}]$ between these…

We give a short review of results on inverse spectral problems for ordinary differential operators on a spatial networks (geometrical graphs). We pay the main attention to the most important nonlinear inverse problems of recovering…

The discrete Schr\"odinger equation with the Dirichlet boundary condition is considered on a half-line lattice when the potential is real valued and compactly supported. The inverse problem of recovery of the potential from the so-called…

In this paper, we study an inverse scattering problem on Liouville surfaces having two asymptotically hyperbolic ends. The main property of Liouville surfaces consists in the complete separability of the Hamilton-Jacobi equations for the…