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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.

This is joint work with Maxime Breden (Ecole Polytechnique, France)

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