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

TL;DR

This paper investigates inverse spectral problems for a class of nonlocal Sturm-Liouville operators with frozen argument, revealing a connection to Chebyshev polynomials and providing a comprehensive description of iso-spectral potentials.

Contribution

It establishes a novel link between the inverse problem's main equation and Chebyshev polynomials, enabling complete characterization of solution uniqueness and iso-spectral potentials.

Findings

Connected the inverse problem to Chebyshev polynomials.

Described all cases of solution uniqueness and degeneracy.

Provided a full description of iso-spectral potentials.

Abstract

The paper deals with Sturm-Liouville-type operators with frozen argument of the form $\ell y:=-y''(x)+q(x)y(a),$ $y^{(\alpha)}(0)=y^{(\beta)}(1)=0,$ where $\alpha,\beta\in\{0,1\}$ 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.