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

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}{