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Standard transmutation operators for the one dimensional Schrödinger operator with a locally integrable potential

Abstract:

We study a special class of operators T satisfying the transmutation relation ([formula presented]−q)Tu=T[formula presented]u 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 represented as a Volterra integral operator of the second kind. We study the boundedness and invertibility properties of the transmutation operators, and use these to obtain a representation for the general distributional solution of the equation [formula presented]−qu=λu, λ∈C, in terms of the general solution of the same equation with λ=0.