Complete Systems of Solutions, Transmutations, and Darboux Transform for Sturm–Liouville Equations in Impedance Form

We present the construction of a complete system of functions associated with the Sturm–Liouville equation in impedance form on a finite interval , given an impedance function . The system, known as the formal powers , is generated through recursive integration of the impedance function and its reciprocal. We establish the completeness of this system in the space with the weight function . Under additional conditions on , we extend this completeness to Sobolev spaces , along with a generalized Taylor formula. We show that the completeness of the formal powers implies key analytic properties for a transmutation operator associated with the Sturm–Liouville equation in impedance form, including the existence of a continuous inverse. Finally, we introduce a formulation of the Darboux‐transformed equation and establish a relation between the transmutation operators to the original and transformed equations.