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 y := −y (x) + q(x)y(a), y (α) (0) = y (β) (1) = 0, where α, β ∈ {0, 1} and a ∈ [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 . 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.

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

“…We note that after publishing the preprint 27 of the present paper, Natalia Bondarenko 28 suggested another interesting and important application of Chebyshev polynomials to inverse spectral problems for operators with frozen argument that deals, however, with a finite-difference approximation of Equation (1).…”

Finite-difference approximation of the inverse Sturm-Liouville problem with frozen argument

“…Inverse problems for the functional-differential equation (1.3) were studied in [7,[21][22][23][24][25][26][27][28][29]. In particular, it has been shown in [23] that the spectrum of the problem (1.3)- (1.4) is the countable set of the eigenvalues {λ n } ∞ n=1 (counting with multiplicities) with the asymptotics…”

“…Degenerate case (see [23,24,26,29]). If π a ∈ Q, that is, π a = j k , j, k ∈ N, j < k, then a part of the spectrum degenerates:…”

“…The crucial step of the reconstruction of the coefficients {q j } is finding the coordinates of some characteristic polynomials with respect to the basis of the Chebyshev polynomials. Note that the continuous problem (1.3)-(1.4) with frozen argument also has a deep relationship with the Chebyshev polynomials (see [29]).…”