Two-dimensional diffusion orthogonal polynomials ordered by a weighted degree

TL;DR

This paper characterizes 2D diffusion operators with polynomial eigenfunctions ordered by weighted degree, extending previous work from standard to arbitrary positive weights.

Contribution

It generalizes the classification of 2D diffusion operators with polynomial eigenfunctions to include arbitrary positive weights in the degree ordering.

Findings

Complete classification of 2D diffusion operators with polynomial eigenfunctions for weighted degrees.

Extension of previous results from standard degree to weighted degree case.

Identification of conditions on the metric and measure for polynomial eigenbasis existence.

Abstract

We study the following problem: describe the triplets $(\Omega,g,\mu)$, $\mu=\rho\,dx$, where $g= (g^{ij}(x))$ is the (co)metric associated with the symmetric second order differential operator $L (f) = \frac{1}{\rho}\sum_{ij} \partial_i (g^{ij} \rho \partial_j f)$ defined on a domain $\Omega$ of $\mathbb R^d$ and such that there exists an orthonormal basis of $\mathcal L^2(\mu)$ made of polynomials which are eigenvectors of $L$, where the polynomials are ranked according to some weighted degree. In a joint paper with D. Bakry and M. Zani this problem was solved in dimension 2 for the usual degree. In the present paper we solve it still in dimension 2, but for a weighted degree with arbitrary positive weights.