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arXiv · 2606.21589

Ambarzumian-type theorems for Hermitian matrices with applications

Matthias Hofmann·Joachim Kerner
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Abstract

A foundational result in inverse spectral theory due to Ambarzumian (1929) states that the Neumann Laplacian on an interval is not isospectral to the Neumann Laplacian with an additional non-zero potential. In this note, our aim is to investigate Ambarzumian-type theorems for certain classes of Hermitian matrices, including well-known matrices such as the discrete Laplacian on finite graphs. In addition, using different methods, we establish an Ambarzumian-type theorem for matrices with vanishing diagonal, in particular, the adjacency matrix on finite graphs. In this way, we generalize existing results on Ambarzumian-type theorems to general finite discrete graphs.

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Matthias Hofmann, Joachim Kerner. 2026-06-19. Ambarzumian-type theorems for Hermitian matrices with applications. https://arxiv.org/abs/2606.21589

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