Analytic approximation of transmutation operators and related systems of functions

摘要

In arXiv:1306.2914 a method for approximate solution of Sturm-Liouvilleequations and related spectral problems was presented based on the constructionof the Delsarte transmutation operators. The problem of numerical approximationof solutions was reduced to approximation of a primitive of the potential by afinite linear combination of certain specially constructed functions obtainedfrom the generalized wave polynomials introduced in arXiv:1208.5984 andarXiv:1208.6166. The method allows one to compute both lower and highereigendata with an extreme accuracy. Since the solution of the approximationproblem is the main step in the application of the method, the properties ofthe system of functions involved are of primary interest. In arXiv:1306.2914two basic properties were established: the completeness in appropriatefunctional spaces and the linear independence. In this paper we present aconsiderably more complete study of the systems of functions. We establishtheir relation with another linear differential second-order equation, find outcertain operations (in a sense, generalized derivatives and antiderivatives)which allow us to generate the next such function from a previous one. Weobtain the uniqueness of the coefficients of expansions in terms of suchfunctions and a corresponding generalized Taylor theorem. We construct theinvertible integral operators transforming powers of the independent variableinto the functions under consideration and establish their commutationrelations with differential operators. We present some error bounds for thesolution of the approximation problem depending on the smoothness of thepotential and show that these error bounds are close to optimal in order. Also,we provide a rigorous justification of the alternative formulation of theproposed method allowing one to make use of the known initial values of thesolutions at an endpoint.