On an extension of the theorem of V. A. Ambarzumyan

Abstract

SynopsisThe Ambarzumyan theorem connecting the Sturm–Liouville problem and the corresponding problem associated with the Fourier differential equation is extended to a class of second order matrix differential systems.

Extension of Ambarzumyan's Theorem to General Boundary Conditions

“…The theorem may be viewed as the first theorem in the history of inverse spectral theory. Recently, Shen and one of us (H. H. Chern), using ideas from [3], proved a vectorial version of Ambarzumyan's theory [4].…”

On a theorem of Ambarzumian

“…attains its minimum at the rst eigenfunction and the minimal value is the rst eigenvalue. Based on this idea, Chakravarty and Acharyya [3] extended the Ambarzumian's theorem to a 2 £ 2 matrix-valued potential. Recently, Chern and Shen [4] proved the more general statement with an n £ n matrix potential.…”

The inverse nodal problem and the Ambarzumyan problem for the p-Laplacian

“…Recently, Chakravarty and Acharyya [6] generalized it to a 2 × 2 vectorial Sturm-Liouville system. In 1997, Chern and Shen [8] extended it to any n-dimensional vectorial Sturm-Liouville system.…”

“…(ii) If the Neumann eigenvalues of (1.1) are given by Part (ii) above is called the Ambarzumyan theorem for the classical Sturm-Liouville operator. Recently, Chakravarty and Acharyya [6] generalized it to a 2 × 2 vectorial Sturm-Liouville system. In 1997, Chern and Shen [8] extended it to any n-dimensional vectorial Sturm-Liouville system.…”