Standard transmutation operators for the one dimensional Schr\"odinger operator with a locally integrable potential

TL;DR

This paper investigates a class of transmutation operators for the one-dimensional Schrödinger operator with locally integrable potentials, providing methods for their construction, properties, and applications to solutions of related differential equations.

Contribution

It introduces a method to construct fundamental transmutation operators from a single operator and characterizes their properties and applications to Schrödinger equations.

Findings

Constructed fundamental transmutation operators from a single operator

Represented solutions of Schrödinger equations via transmutation operators

Analyzed boundedness and invertibility of these operators

Abstract

We study a special class of operators T satisfying the transmutation relation (Tu)"-qTu=Tu" in the sense of distributions, where q is a locally integrable function, and u belongs to a suitable space of distributions depending on the smoothness properties of q. A method which allows one to construct a fundamental set of transmutation operators of this class in terms of a single particular transmutation operator is presented. Moreover, following [27], we show that a particular transmutation operator can be realized as a Volterra integral operator of the second kind. We study the boundedness and invertibility properties of the transmutation operators, and used it to obtain a representation for the general distributional solution of the equation u"-qu=zu, in terms of the general solution of the same equation with z=0.