Solutions of differential equations in Freud-weighted Sobolev spaces
Numerical Analysis
2026-02-11 v1 Numerical Analysis
Analysis of PDEs
Classical Analysis and ODEs
Functional Analysis
Abstract
We lay some mathematically rigorous foundations for the resolution of differential equations with respect to semi-classical bases and topologies, namely Freud-Sobolev polynomials and spaces. In this quest, we uncover an elegant theory melding various topics in Numerical and Functional Analysis: Poincar\'e inequalities, Sobolev orthogonal polynomials, Painlev\'e equations and more. Brought together, these ingredients allow us to quantify the compactness of Sobolev embeddings on Freud-weighted spaces and finally resolve some differential equations in this topology. As an application, we rigorously and tightly enclose solutions of the Gross-Pitaevskii equation with sextic potential.
Keywords
Cite
@article{arxiv.2501.13672,
title = {Solutions of differential equations in Freud-weighted Sobolev spaces},
author = {Maxime Breden and Hugo Chu},
journal= {arXiv preprint arXiv:2501.13672},
year = {2026}
}