Solutions of differential equations in Freud-weighted Sobolev spaces

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: Poincaré inequalities, Sobolev orthogonal polynomials, Painlevé 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.

💡 Research Summary

The paper establishes a rigorous framework for solving differential equations in Sobolev spaces equipped with Freud‑type exponential weights. Starting from a probability measure (\nu(dx)=e^{-V(x)}dx/Z) with a smooth even potential (V), the authors recall that the associated orthogonal polynomials ({p_n}) form an orthonormal basis of (L^2(\nu)). However, for non‑classical weights the differentiation operator (\partial_x) is not diagonal in this basis, which hampers spectral methods. To overcome this, the authors introduce a natural inner product on the weighted Sobolev space (H^1(\nu)): \