2016/08/23 by Hugo M. Campos, Campos, Hugo M · 2 citations
Mathematics · Economics, Econometrics and Finance · #Differential Equations and Boundary Problems #Stochastic processes and financial applications #Advanced Mathematical Physics Problems
paper · pdf · doi:10.48550/arxiv.1608.06573
We study a special class of operators T satisfying the transmutation relation\n(Tu)"-qTu=Tu" in the sense of distributions, where q is a locally integrable\nfunction, and u belongs to a suitable space of distributions depending on the\nsmoothness properties of q. A method which allows one to construct a\nfundamental set of transmutation operators of this class in terms of a single\nparticular transmutation operator is presented. Moreover, following [27], we\nshow that a particular transmutation operator can be realized as a Volterra\nintegral operator of the second kind. We study the boundedness and\ninvertibility properties of the transmutation operators, and used it to obtain\na representation for the general distributional solution of the equationν"-qu=zu, in terms of the general solution of the same equation with z=0.\n