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

Abstract

This paper deals with the discrete system being the finite-difference approximation of the Sturm-Liouville problem with frozen argument. The inverse problem theory is developed for this discrete system. We describe the two principal cases: degenerate and non-degenerate. For these two cases, appropriate inverse problems statements are provided, uniqueness theorems are proved, and reconstruction algorithms are obtained. Moreover, the relationship between the eigenvalues of the continuous problem and its finite-difference approximation is investigated. We obtain the "correction terms" for approximation of the discrete problem eigenvalues by using the eigenvalues of the continuous problem. Relying on these results, we develop a numerical algorithm for recovering the potential of the Sturm-Liouville operator with frozen argument from a finite set of eigenvalues. The effectiveness of this algorithm is illustrated by numerical examples.

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

“…Hence, (28) holds for all 𝜈 ≥ 0. Substituting (28) into the first formula of this proof, we arrive at…”

“…Moreover, by virtue of (45) along with Proposition 1 and Lemma 2, the kernel of A (1,𝛽) 𝑗,k for 𝛽 ∈ {0, 1} is a linear hull of the vector X = X 0 determined by (58) for 𝛼 = 1 and the corresponding 𝛽 as well as z 0 = 0 since 𝑗 is odd in both subcases (III) and (IV) and, hence, T 𝑗 (0) = 0. Thus, formulae (61), (63), and (64) for components of X can be easily obtained using (11), (28), and (58). Now let (𝛼, 𝛽) = (0, 1).…”

“…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).…”