Tzong-Mo Tsai | Searcharxiv

arXiv2025

Partial Inverse Spectral Problems for Sturm-Liouville Operators with Frozen Arguments on a Star-Shaped Graph

This paper is devoted to the study of a partial inverse spectral problem for Sturm-Liouville operators with frozen arguments on a star-shaped graph. The potentials are assumed to be known a priori on all edges except one, and the objective is to reconstruct the unknown potential on the remaining edge using a subset of the spectral data. A constructive algorithm for solving this problem is presented, which relies on the Riesz basis property of a system of vector functions.

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arXiv2024

Inverse spectral problem of Sturm-Liouville equation with many frozen arguments

This research was devoted to investigate the inverse spectral problem of Sturm-Liouville operator with many frozen arguments. Under some assumptions, the authors obtained uniqueness theorems. At the end, a numerical simulation for the inverse problem was presented.

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arXiv2021

Sturm-Liouville-type operators with frozen argument and Chebyshev polynomials

The paper deals with Sturm-Liouville-type operators with frozen argument of the form $\ell y:=-y''(x)+q(x)y(a),$ $y^{(α)}(0)=y^{(β)}(1)=0,$ where $α,β\in\{0,1\}$ and $a\in[0,1]$ is an arbitrary fixed rational number. Such nonlocal operators belong to the so-called loaded differential operators, which often appear in mathematical physics. We focus on the inverse problem of recovering the potential $q(x)$ from the spectrum of the operator $\ell.$ Our goal is two-fold. Firstly, we establish a deep connection between the so-called main equation of this inverse problem and Chebyshev polynomials of the first and the second kinds. This connection gives a new perspective method for solving the inverse problem. In particular, it allows one to completely describe all non-degenerate and degenerate cases, i.e. when the solution of the inverse problem is unique or not, respectively. Secondly, we give a complete and convenient description of iso-spectral potentials in the space of complex-valued integrable functions.

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