Inverse problems for Jacobi operators with mixed spectral data

We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.

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“…[8,12,13,15,20] and references therein) and discrete (see e.g. [2][3][4][5][6][7][8][9][10][11]16,18] and references therein) settings.…”

Section: Introductionmentioning

confidence: 99%

Ambarzumian-type Problems for Discrete Schrödinger Operators

Hatinoğlu

Eakins

Frendreiss

et al. 2021

Complex Anal. Oper. Theory

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On Meromorphic Inner Functions in the Upper Half-Plane

Hati̇noğlu

2023

Fields Institute Communications

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An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod

Mirzaei,

Abbasnavaz,

Ghanbari

2024

Computational Methods in Applied Mathematics

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The free longitudinal vibrations of a rod are described by a differential equation of the form ( P ( x ) y ′ ) ′ + λ P ( x ) y ( x ) = 0 {(P(x)y\prime)^{\prime}+\lambda P(x)y(x)=0} , where P ⁢ ( x ) {P(x)} is the cross section area at point x and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form 𝐀 ⁢ Y = Λ ⁢ 𝐁 ⁢ Y {\mathbf{A}Y=\Lambda\mathbf{B}Y} , where 𝐀 {\mathbf{A}} and 𝐁 {\mathbf{B}} are Jacobi and diagonal matrices dependent to cross section P ⁢ ( x ) {P(x)} , respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section P ⁢ ( x ) {P(x)} by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is O ⁢ ( h 2 ) {O(h^{2})} .

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Inverse problems for Jacobi operators with mixed spectral data

Cited by 3 publications

References 24 publications

Ambarzumian-type Problems for Discrete Schrödinger Operators

Ambarzumian-type Problems for Discrete Schrödinger Operators

On Meromorphic Inner Functions in the Upper Half-Plane

An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod

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