Transmutation operators for the Sturm--Liouville equation in impedance form with an integrable potential

Abstract

In this work, we study a family of transmutation operators for the Sturm--Liouville operator in impedance form with an integrable potential satisfying a smallness condition. Specifically, we consider transmutations associated with the pair \[\textbf{L}_r=\frac{1}{r^2 (x)}\frac{d}{dx}r^2 (x)\frac{d}{dx} \quad \text{and} \quad \textbf{M}=\frac{d^2}{dx^2}. \] We extend existing results by showing that a particular transmutation operator on $W^{3,1} (-a,a)$ can be represented in integral form, where the kernel function satisfies a corresponding Goursat problem. We establish the existence of this kernel using two distinct approaches: the classical method of successive approximations and an application of Picard's theorem.