On Ambarzumyan-type Inverse Problems of Vibrating String Equations

2020/08/29 by Yuri Ashrafyan, Dominik L. Michels, Ashrafyan, Yuri +1
Engineering · Mathematics · #34A55 #34L15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods in engineering #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2008.13035

openalex publication_date 2020/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the inverse spectral theory of vibrating string equations. In this regard, first eigenvalue Ambarzumyan-type uniqueness theorems are stated and proved subject to separated, self-adjoint boundary conditions. More precisely, it is shown that there is a curve in the boundary parameters' domain on which no analog of it is possible. Necessary conditions of the n-th eigenvalue are identified, which allows to state the theorems. In addition, several properties of the first eigenvalue are examined. Lower and upper bounds are identified, and the areas are described in the boundary parameters' domain on which the sign of the first eigenvalue remains unchanged. This paper contributes to inverse spectral theory as well as to direct spectral theory.

Related

• An Extension of The First Eigen-type Ambarzumyan theorem
• A new Improvement of Ambarzumyan's Theorem
• On the Well-possedness of the Problem of Reconstruction of Non-separate Boundary Conditions
• Eigenvalue ratios for vibrating String equations with concave densities
• Direct and inverse problems for a third order self-adjoint differential operator with periodic boundary conditions and nonlocal potential