Complete systems of solutions, transmutations and Darboux transform for Sturm-Liouville equations in impedance form

TL;DR

This paper constructs a complete system of functions for Sturm-Liouville equations in impedance form, demonstrating their completeness in various function spaces and exploring related transmutation and Darboux transformations.

Contribution

It introduces formal powers generated by recursive integration of the impedance function, proving their completeness and linking them to transmutation operators and Darboux transformations.

Findings

Complete system of functions (formal powers) constructed and proven complete in $L^p$ and Sobolev spaces.

Established existence and properties of transmutation operators with continuous inverses.

Linked Darboux transformations to transmutation operators for impedance form Sturm-Liouville equations.

Abstract

We present the construction of a complete system of functions associated with the Sturm-Liouville equation in impedance form on a finite interval $I$, given an impedance function $a\in L^2(I)$. The system, known as the formal powers, is generated through recursive integration of the impedance function $a$ and its reciprocal. We establish the completeness of this system in the space $L^p$ with the weight function $a^2$. Under additional conditions on $a$, we extend the completeness of this completeness to Sobolev spaces $W^{1,p}(I)$, 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.