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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 integrablefunction, and u belongs to a suitable space of distributions depending on thesmoothness properties of q. A method which allows one to construct afundamental set of transmutation operators of this class in terms of a singleparticular transmutation operator is presented. Moreover, following [27], weshow that a particular transmutation operator can be realized as a Volterraintegral operator of the second kind. We study the boundedness andinvertibility properties of the transmutation operators, and used it to obtaina representation for the general distributional solution of the equationu"-qu=zu, in terms of the general solution of the same equation with z=0.

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