Representation of solutions to the one-dimensional Schr\"odinger equation in terms of Neumann series of Bessel functions

TL;DR

This paper introduces a novel Neumann series of Bessel functions representation for solutions of the 1D Schrödinger equation, enabling efficient spectral problem approximations and eigendata computations.

Contribution

It develops a new solution representation using transmutation operators, improving uniform approximation and numerical computation of spectral data.

Findings

Partial sums approximate solutions uniformly in mbda

Method enables computation of large eigendata sets with stable accuracy

Representation facilitates efficient spectral problem solving

Abstract

A new representation of solutions to the equation $-y"+q(x)y=\omega^2 y$ is obtained. For every $x$ the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter $\omega$. Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to $\omega$ which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.