Inverse Spectral Problem Research Articles

Inverse Spectral Problem for a Star Graph of Stieltjes Strings with Given Numbers of Masses on the Edges

Inverse Spectral Problem for a Star Graph of Stieltjes Strings with Given Numbers of Masses on the Edges

Inverse problems of recovering first-order integro-differential operators from spectra

Inverse spectral problems are studied for first-order integro-differential operators on a finite interval. These problems consist in recovering some components of the kernel from one or multiple spectra. Uniqueness theorems are proved for this class of inverse problems.

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 validate the theoretical convergence. We also consider the inverse spectral problem of estimating/recovering the refractive index from the knowledge of the Steklov eigenvalues, since the eigenvalues are monotone with respect to a real-valued refractive index implying that they can be used for nondestructive testing. Some numerical examples are provided for the inverse spectral problem.

Inverse spectral problems for Dirac operators on a star graph with mixed boundary conditions

This paper is devoted to some inverse spectral problems for Dirac operators on a star graph with mixed boundary conditions in boundary vertices. By making use of Rouché's theorem, we derive the eigenvalue asymptotics of these operators. Besides, we show that for each of these operators, if the potentials are known a priori for all but one edge on the graph, then the potential on the remaining edge is uniquely determined by part of the potential on this edge and part of its spectrum. Our method relies upon some estimates of infinite products given by Horváth.

Direct and Inverse Spectral Problems for Rank-One Perturbations of Self-adjoint Operators

For a given self-adjoint operator A with discrete spectrum, we completely characterise possible eigenvalues of its rank-one perturbations B and discuss the inverse problem of reconstructing B from its spectrum.

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness, and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.

ON THE INTEGRATION OF A HIRERARCHY FOR THE PEREODIC TODA LATTICE WITH A FREE TERM

In this article, we have explored the Toda lattice hierarchy in the class of periodic functions with a free term. We have given an effective method of constructing of the periodic Toda lattice hierarchy with a free term. We have discussed the complete integrability of the constructed systems that is based on the inverse spectral problem of an associated discrete Hill`s equation with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy.

Mixed data in inverse spectral problems for the Schrödinger operators

We consider the Schrödinger operator on a finite interval with an L^1 -potential. We prove that the potential can be uniquely recovered from one spectrum and subsets of another spectrum and point masses of the spectral measure (or norming constants) corresponding to the first spectrum. We also solve this Borg–Marchenko-type problem under some conditions on two spectra, when missing part of the second spectrum and known point masses of the spectral measure have different index sets.

Inverse spectral problems for Hill-type operators with frozen argument

The paper deals with nonlocal differential operators possessing a term with frozen (fixed) argument appearing, in particular, in modelling various physical systems with feedback. The presence of a feedback means that the external affect on the system depends on its current state. If this state is taken into account only at some fixed physical point, then mathematically this corresponds to an operator with frozen argument. In the present paper, we consider the operator $$Ly\equiv -y^{\prime \prime }(x)+q(x)y(a),$$ $$y^{(\nu )}(0)=\gamma y^{(\nu )}(1),$$ $$\nu =0,1,$$ where $$\gamma \in {\mathbb C}{\setminus }\{0\}.$$ The operator L is a nonlocal analog of the classical Hill operator describing various processes in cyclic or periodic media. We study two inverse problems of recovering the complex-valued square-integrable potential q(x) from some spectral information about L. The first problem involves only single spectrum as the input data. We obtain complete characterization of the spectrum and prove that its specification determines q(x) uniquely if and only if $$\gamma \ne \pm 1.$$ For the rest (periodic and antiperiodic) cases, we describe classes of iso-spectral potentials and provide restrictions under which the uniqueness holds. The second inverse problem deals with recovering q(x) from the two spectra related to $$\gamma =\pm 1.$$ We obtain necessary and sufficient conditions for its solvability and establish that uniqueness holds if and only if $$a=0,1.$$ For $$a\in (0,1),$$ we describe classes of iso-bispectral potentials and give restrictions under which the uniqueness resumes. Algorithms for solving both inverse problems are provided. In the appendix, we prove Riesz-basisness of an auxiliary two-sided sequence of sines.

Spectral properties of the singular Friedrichs–Lee Hamiltonian

We show that the Friedrichs–Lee model, which describes the one-excitation sector of a two-level atom interacting with a structured boson field, can be generalized to singular atom–field couplings. We provide a characterization of its spectrum and resonances and discuss the inverse spectral problem.

Direct and inverse spectral problems for a Dirac operator with non-local potential

For a Dirac operator with non-local potential on a finite segment, a method of reconstruction of non-local potential from the spectral data is developed. Description of spectral data for such class of operators is given.

Solving An Inverse Problem For The Sturm-Liouville Operator With A Singular Potential By Yurko's Method

An inverse spectral problem for the Sturm-Liouville operator with a singular potential from the class $W_2^{-1}$ is solved by the method of spectral mappings. We prove the uniqueness theorem, develop a constructive algorithm for solution and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and the non-self-adjoint cases.

Inverse spectral problems for the Dirac operator with complex-valued weight and discontinuity

In this paper, we consider inverse problems for the Dirac operator with complex-valued weight and the jump conditions inside the interval. We prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra.

Inverse spectral problem for Jacobi operators and Miura transformation

Abstract We study a Miura-type transformation between Kac - van Moerbeke (Volterra) and Toda lattices in terms of the inverse spectral problem for Jacobi operators, which appear in the Lax representation for such systems. This inverse problem method, which amounts to reconstruction of the operator from the moments of its Weyl function, can be used in solving initial-boundary value problem for both systems. It is shown that the Miura transformation can be easily described in terms of these moments. Using this description we establish a bijection between the Volterra lattices and the class of Toda lattices which is characterized by positivity of Jacobi operators in their Lax representation. Also, we discuss an implication of the latter result to the spectral theory.

Algorithm for Numerical Solution of Inverse Spectral Problems Generated by Sturm-Liouville Operators of an Arbitrary Even Order

Algorithm for Numerical Solution of Inverse Spectral Problems Generated by Sturm-Liouville Operators of an Arbitrary Even Order

Adaptative regularization parameter for poisson noise with a bilevel approach: application to spectral computerized tomography

In this paper, we present a method of choice of an adaptative regularization parameter for data corrupted by Poisson noise based on a bilevel approach. The forward operator considered is nonlinear. The existence and unicity of the smoothed lower level problem, the differentiability properties of the constraint, and the adjoint method used to calculate the gradient of the reduced functional are studied in detail. The variance of the KL functional for Poisson noise is also investigated. The method is applied to the spectral CT inverse problem. Better reconstruction results are obtained with the bilevel method of choice than with a scalar regularization parameter.

A half‐inverse problem for singular diffusion operator with certain boundary conditions

AbstractIn this paper, we studied the half inverse spectral problem for singular diffusion operator with certain boundary conditions. The discontinuity function in this operator is defined as and α > 0, α ≠ 1, β > 0, β ≠ 1 and a1, a2 ∈ (0, π), , . We prove that the potential functions p(x) and q(x) are determined uniquely by using the Yang–Zettl and Hocstadt–Lieberman methods. We examine that if potential functions q(x) and p(x) are prescribed over the interval , then reconstruction of the potential functions q(x) and p(x) by one spectrum on the (0, π).

Direct and Inverse Spectral Problems in the Theory of Oscillations of Elastic Plates with Additional Point Interactions

This paper is devoted to a new statement and the study of direct and inverse spectral problems for small linear oscillations of orthotropic plates that carry concentrated masses at a finite set of points, which, in turn, are connected to a stationary base by elastic springs with known stiffness coefficients.

Spectral data characterization for the Sturm–Liouville operator on the star-shaped graph

The inverse spectral problems are studied for the Sturm–Liouville operator on the star-shaped graph and for the matrix Sturm–Liouville operator with one boundary condition in the general self-adjoint form. We obtain necessary and sufficient conditions of solvability for these two inverse problems, and also prove their local solvability and stability.

Uniqueness Theorems for Inverse Problems of Discontinuous Sturm–Liouville Operator

In this paper, inverse spectral problems of discontinuous Sturm–Liouville problems contained in the discontinuous coefficient and discontinuity conditions at an interior point of the finite interval are, respectively, studied according to