Inverse resonance problem with partial information on the interval

Periodic Jacobi operator with finitely supported perturbations: The inverse resonance problem

We study an inverse resonance problem on the line in which we aim at determining a compactly supported and integrable perturbation of a fixed Pöschl-Teller potential. We define the resonances as the poles of the reflection coefficients with a negative imaginary part. Given the zeros and the poles of one of the reflection coefficients, we are able to determine uniquely the perturbation of a Pöschl-Teller potential when its support is to the left or to the right of zero on the line and also in the remaining cases under the addition of other hypothesis or extra spectral data. We also give asymptotics of the resonances and show that they are asymptotically located on two logarithmic branches.

We consider the class of CMV operators with super-exponentially decaying Verblunsky coefficients. For these we define the concept of a resonance. Then we prove the existence of Jost solutions and a uniqueness theorem for the inverse resonance problem: given the location of all resonances, taking multiplicities into account, the Verblunsky coefficients are uniquely determined.

Stabilized Numerical Analytic Prolongation with Poles

This work investigates both direct and inverse problems of the variable-exponent sub-diffusion model, which attracts increasing attentions in both practical applications and theoretical aspects. Based on the perturbation method, which transfers the original model to an equivalent but more tractable form, the uniqueness of solutions to the variable-exponent subdiffusion equation with homogeneous Dirichlet boundary condition are established from its partial information, which results in the uniqueness of the inverse space-dependent source problem from local internal observation or partial boundary flux data. Then, based on the variational identity connecting the inversion input data with the unknown source function, we propose a weak norm and prove the conditional stability for the inverse problem in this norm. The iterative thresholding algorithm and Nesterov iteration scheme are employed to numerically reconstruct the smooth and non-smooth sources, respectively. Numerical experiments are performed to investigate their effectiveness.

The inverse spectral problem for the Dirac operators defined on the interval [0, π] with self-adjoint separated boundary conditions is considered. Some uniqueness results are obtained, which imply that the pair of potentials (p(x), r(x)) and a boundary condition are uniquely determined even if only partial information is given on (p(x), r(x)) together with partial information on the spectral data, consisting of either one full spectrum and a subset of norming constants, or a subset of pairs of eigenvalues and the corresponding norming constants. Moreover, the authors are also concerned with the situation where both p(x) and r(x) are Cn-smoothness at some given point.

In this paper, we study the inverse spectral problems for Sturm–Liouville equations with boundary conditions linearly dependent on the spectral parameter and show that the potential of such problem can be uniquely determined from partial information on the potential and parts of two spectra, or alternatively, from partial information on the potential and a subset of pairs of eigenvalues and the normalization constants of the corresponding eigenvalues.

On regularization results for a two-dimensional nonlinear time-fractional inverse diffusion problem

We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a one-dimensional Schrodinger operator H = -(d^(2)/(dx^(2)) + q determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of H on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl m-function techniques and densities of zeros of a class of entire functions.

We consider an inverse spectral problem for a string equation, subject to the self-adjoint boundary conditions and with partial information given on the density function p(x) which is assumed to be a positive piecewise continuously differentiable function defined in the interval [0, 1]. We investigate the situation where (i) the density p is known only on the subinterval [a, 1] for some a ∊ (0, 1) and (ii) only part of one spectrum or two spectra of the problem is given. We will, under suitable circumstances, prove that (i) and (ii) are sufficient to uniquely determine the values of the density function p on the entire interval [0, 1].

The inverse matrix eigenvalue problem is the problem of reconstruction of a matrix (determination of the unknown parameters of a certain matrix) for a given finite set of spectral data (points of the spectrum). The spectral data used in this case may contain complete or only partial information on the eigenvalues or eigenvectors. The aim of an inverse eigenvalue problem is to construct a matrix that preserves a certain specific structure and the prescribed spectral properties. There are two basic questions connected with any inverse eigenvalue problem: the theoretical question on the possibility of construction of its solutions and the practical question of its determination. The main investigations connected with solvability deal with establishing necessary or sufficient conditions under which the inverse eigenvalue problem is solvable. On the other hand, the main difficulty in calculations is connected with the development of a numerical procedure which would enable us to construct the matrix itself according to the a-priori specified spectral data. Both questions are important and complicated. Numerous works are devoted to the conditions of existence and uniqueness of solutions for different formulations of inverse matrix spectral problems and the methods used for their solution (see, e.g., [1–3, 7, 9–13, 15] and the literature therein). In the present work, we consider a numerical algorithm used for the solution of inverse eigenvalue problems under the assumption that this solution exists. Hence, we consider inverse algebraic eigenvalue problems.

In this paper, we consider the spectral problem for a Stieltjes string with both ends fixed and with one-dimensional damping at an interior point. We solve an inverse problem of recovering parameters of the Stieltjes string using partial information on the string and partial information on the spectrum. First we prove uniqueness and give an algorithm of recovering the unknown parameters of the string by virtue of a finite number of distinct not pure imaginary complex eigenvalues and a certain even number of real eigenvalues. For the case where only complex eigenvalues are used we solve the existence problem, i.e. propose conditions on a set of complex numbers necessary and sufficient for these numbers to be a subset of the spectrum of the problem.

Numerical Solution of an Inverse Multifrequency Problem in Scalar Acoustics

This paper and a companion paper by Bal (Bal G 2000 Inverse Problems 16 1013) are parts I and II of a series dealing with thereconstruction from boundary measurements of the scatteringoperator of homogeneous linear transport equations. This firstpart addresses the one-dimensional half-space setting, for both time-dependent and steady-state problems. We propose areconstruction procedure based on a Riccati equation for theresponse operator. In the steady-state case, we show that thereconstruction is a well-posed inverse problem provided thescattering operator satisfies enough symmetry constraints. Theproblem becomes ill posed when some of these constraints aredropped. We show that ill-posed problems in the steady-statecase become well posed in the time-dependent setting with time-independent scattering operators.

We consider the inverse spectral problem for a Sturm-Liouville problem on the unit interval $[0,1]$ . We obtain some uniqueness results, which imply that the potential q can be completely determined even if only partial information is given on q together with partial information on the spectral data, consisting of the spectrum and normalizing constants. Moveover, we also investigate the problem of missing both eigenvalues and normalizing constants in the situation where the potential q is $C^{2k-1}$ near a suitable point.