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

Abstract

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

18 pages, 5 figures

References in corpus (1)

• Inverse spectral problems for Hill-type operators with frozen argument

Cited by in corpus (2)

• Uniform stability of recovering the Sturm-Liouville operators with frozen argument
• Inverse problem for a differential operator on a star-shaped graph with nonlocal matching condition