Construction of a transmutation for the one-dimensional Schr\"odinger operator and a representation for solutions

TL;DR

This paper introduces a novel series representation for solutions of the one-dimensional Schrödinger equation with a key property that the truncation error remains constant regardless of the spectral parameter, facilitating computational applications.

Contribution

The authors develop a new series solution with w-independent truncation error and simple recurrence formulas for coefficients, enhancing computational efficiency for the Schrödinger equation.

Findings

Series representation with w-independent truncation error

Simple recurrent formulas for coefficients

Applicable for efficient numerical computation

Abstract

A new representation for solutions of the one-dimensional Schr\"odinger equation -u"+q(x)u=w^2u is obtained in the form of a series possessing the following attractive feature. The truncation error is w-independent for all real w. For the coefficients of the series simple recurrent integration formulas are obtained which make the new representation applicable for computation.