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

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.