On the Canonical Solution of the Sturm–Liouville Problem with Singularity and Turning Point of Even Order

Abstract

Abstract. In this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm-Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm-Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm-Liouville equation with turning point to the equation with singularity, then we study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem.

On the uniqueness of the solution of dual equation of a singular Sturm–Liouville problem

“…From Theorem 4 of [8] we have the following theorem: (8), then for t ∈ ω s , s = 0, 1, (ρ, t) ∈ , |ρ| → ∞, Im ρ ≥ 0, m = 0, 1, the following estimate hold:…”

“…From [8], we know that for t ∈ (0, T )\{t 1 }, the following asymptotic distribution for each {λ n } n≥1 holds:…”

“…The representing solution of the infinite product form plays an important role for investigating the corresponding dual equations. Note that, for eigenvalue approximation and infinite product representation of solutions in a singularity and a turning point case, see [8]. The inverse problem with a singularity was also studied by the author in [11].…”

“…This paper proposes and tests an extension of a technique, first by indicating the infinite product solutions of the equation with a singularity, and then we derive the associate dual equation by this infinite product form of solutions and prove the uniqueness solution of the dual equation. Note that, in [8] we studied the infinite product representation of solutions in a singularity case.…”