Numerical Analysis of differential equations on weighted Sobolev spaces: beyond classical orthogonal polynomials

We lay mathematical foundations for the Numerical Analysis of differential equations on Sobolev spaces weighted by a Gibbs probability measure $\nu(\mathrm{d} x) = e^{-V(x)}\mathrm{d} x/\mathcal{Z}$ on the real line. Over recent decades, the Functional Analysis of these spaces has been thoroughly developed to study Schr\"odinger-type equations and diffusion processes. While such equations should therefore be amenable to a numerical resolution with respect to orthogonal polynomials, this feat has only ever been achieved with respect to classical bases. We bridge this gap by showing that such equations can be solved with respect to suitable bases by factorising their leading linear component. In particular, we propose a new natural notion of Sobolev orthogonal polynomials, simpler and more tractable than those arising from the usual Sobolev inner product. In the case of $V$ being an even polynomial, we further establish quantitative estimates for the compactness of the embedding $H^1(\nu)\hookrightarrow L^2(\nu)$, uncovering a connection with the growth of the Jacobi recurrence coefficients, which are solutions of corresponding Painlev\'e-type discrete equations. As an application, we rigorously and tightly enclose solutions of the Gross--Pitaevskii equation with sextic potential and rigorously demonstrate the phenomenon of stochastic resonance via a computer-assisted proof.