Orthogonal polynomials and diffusion operators

Abstract

We study the following problem: describe the triplets ( Ω , g , μ ) where g = ( g i j ( x ) ) is the (co)metric associated with the symmetric second order differential operator L ( f ) = 1 ρ ∑ i j ∂ i ( g i j ρ ∂ j f ) defined on a domain Ω of ℝ d (that is L is a diffusion operator with reversible measure μ ( d x ) = ρ ( x ) d x ) and such that there exists an orthonormal basis of ℒ 2 ( μ ) made of polynomials which are at the same time eigenvectors of L , where the polynomials are ranked according to their natural degree. We reduce this problem to a certain algebraic problem (for any d ) and we find all solutions for d = 2 when Ω is compact. Namely, in dimension d = 2 , and up to affine transformations, we find 10 compact domains Ω plus a one-parameter family. The proof that this list is exhaustive relies on the Plücker-like formulas for the projective dual curves applied to the complexification of ∂ Ω . We then describe some geometric origins for these various models. We also give some description of the non-compact cases in this dimension.