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

Abstract

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.

Necessary and sufficient conditions for the spectra of the Sturm–Liouville operators with frozen argument

“…Various aspects of inverse spectral problems for operators with frozen argument were studied in [19][20][21][22][23][24][25][26][27][28][29]. In particular, the inverse problem by the spectrum {λ n } n∈N was investigated in [22-24, 26, 27], wherein the cases of rational and irrational a/π were treated apart by different techniques.…”

Solving an inverse nodal problem with Herglotz–Nevanlinna functions in boundary conditions using the second‐kind Chebyshev wavelets method

“…However, in recent years, some mathematicians have made some important studies on this subject. [17][18][19][20][21][22] In 2007, Babolian and Fattahzadeh 23 considered Chebyshev wavelets method to get approximate solution of integral equations by considering 1st kind Chebyshev polynomials. Akbarpoor et al 24 studied the BVP (1) and (2) with 𝛿 k = 0, k = 1, 2, … , d, Δ l = 0, l = 2, 3, … , D, and h 0 = H 0 = 0 and used FCW and Chebyshev interpolation methods to get solution of inverse nodal problem in 2019.…”

“…There are few studies on numerical solutions to this problem. However, in recent years, some mathematicians have made some important studies on this subject 17–22 . In 2007, Babolian and Fattahzadeh 23 considered Chebyshev wavelets method to get approximate solution of integral equations by considering 1st kind Chebyshev polynomials.…”

An Inverse Sturm–Liouville-Type Problem with Constant Delay and Non-Zero Initial Function

“…from its spectrum, where b ∈ [0, π] and α, β ∈ {0, 1} (see [28][29][30][31][32][33][34][35] and references therein). In particular, its unique solvability depends on the value of b as well as on α and β.…”

“…In particular, its unique solvability depends on the value of b as well as on α and β. We note that both related to Inverse Problem 1 situations: b = 0, α = 1, β = 0 and b = 0, α = β = 1 belong to the so-called non-generate case, when the solution is unique (see, e.g., [28,29,32]). We note that Theorem 2 also formally holds for a = π, which follows from Theorem 4.1 in [28] or Theorem 2 in [29].…”