# A representation of the transmutation kernels for the Schr\"odinger operator in terms of eigenfunctions and applications

## TL;DR

This paper develops improved, uniformly convergent representations of transmutation kernels for the Schrödinger operator using eigenfunctions, with practical numerical applications.

## Contribution

It introduces methods to enhance the convergence of kernel representations for the Schrödinger operator in terms of eigenfunctions, enabling more accurate numerical computations.

## Key findings

- Derived new series representations with improved convergence.
- Provided numerical illustrations demonstrating the effectiveness.
- Achieved uniform and absolute convergence of kernel expansions.

## Abstract

The representations of the kernels of the transmutation operator and of its inverse relating the one-dimensional Schr\"odinger operator with the second derivative are obtained in terms of the eigenfunctions of a corresponding Sturm-Liouville problem. Since both series converge slowly and in general only in a certain distributional sense we find a way to improve these expansions and make them convergent uniformly and absolutely by adding and subtracting corresponding terms. A numerical illustration of the obtained results is given.