Analytic approximation of transmutation operators and related systems of functions

TL;DR

This paper advances the approximation of solutions to Sturm-Liouville problems by analyzing transmutation operators and special function systems, achieving high accuracy and establishing their mathematical properties.

Contribution

It provides a comprehensive study of the function systems used in approximating Sturm-Liouville solutions, including their relations, properties, and error bounds.

Findings

Established completeness and linear independence of the function systems.

Derived error bounds depending on potential smoothness.

Constructed invertible operators linking powers of variables to the functions.

Abstract

In arXiv:1306.2914 a method for approximate solution of Sturm-Liouville equations and related spectral problems was presented based on the construction of the Delsarte transmutation operators. The problem of numerical approximation of solutions was reduced to approximation of a primitive of the potential by a finite linear combination of certain specially constructed functions obtained from the generalized wave polynomials introduced in arXiv:1208.5984 and arXiv:1208.6166. The method allows one to compute both lower and higher eigendata with an extreme accuracy. Since the solution of the approximation problem is the main step in the application of the method, the properties of the system of functions involved are of primary interest. In arXiv:1306.2914 two basic properties were established: the completeness in appropriate functional spaces and the linear independence. In this paper we present a considerably more complete study of the systems of functions. We establish their relation with another linear differential second-order equation, find out certain operations (in a sense, generalized derivatives and antiderivatives) which allow us to generate the next such function from a previous one. We obtain the uniqueness of the coefficients of expansions in terms of such functions and a corresponding generalized Taylor theorem. We construct the invertible integral operators transforming powers of the independent variable into the functions under consideration and establish their commutation relations with differential operators. We present some error bounds for the solution of the approximation problem depending on the smoothness of the potential and show that these error bounds are close to optimal in order. Also, we provide a rigorous justification of the alternative formulation of the proposed method allowing one to make use of the known initial values of the solutions at an endpoint.