Sturm–Liouville‐type operators with frozen argument and Chebyshev polynomials

• Published in: Mathematical methods in the applied sciences 2022-11, Vol.45 (16), p.9635-9652
• Main Authors: Tsai, Tzong‐Mo, Liu, Hsiao‐Fan, Buterin, Sergey, Chen, Lung‐Hui, Shieh, Chung‐Tsun
• Format: Article
• Language: English
• Subjects: Chebyshev approximation; Chebyshev polynomials; Differential equations; frozen argument; functional‐differential operator; Inverse problems; inverse spectral problem; isospectral potentials; Jacobi matrices; loaded differential equation; Operators (mathematics); Polynomials; Spectral theory
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8327

In recent years, there appeared a growing interest in the inverse spectral theory for functional‐differential operators with frozen argument. Such operators are nonlocal and belong to the so‐called loaded differential operators, which frequently appear in mathematics as well as natural sciences and engineering. However, to various classes of nonlocal operators, classical methods of the inverse spectral theory are not applicable. For this reason, it is relevant to develop new methods and approaches for solving inverse problems for operators of this type. We establish a deep connection between the inverse problem for operators with frozen argument and Chebyshev polynomials of the first and the second kinds. Appearing to be of an interest from the point of view of the Chebyshev polynomials theory itself, this connection gives a new perspective method for studying inverse problems for operators with frozen argument. In particular, it allows one to completely describe all non‐degenerate and degenerate cases, that is, when the corresponding inverse problem is uniquely solvable or not, respectively. Moreover, it gives a convenient description of all isospectral potentials in the degenerate case, which is demonstrated by some illustrative examples.