The CMV bispectral problem

TL;DR

This paper solves a bispectral problem for orthogonal Laurent polynomials on the unit circle, showing that only the Lebesgue measure produces eigenfunction sequences for differential operators, extending classical results to a new setting.

Contribution

It generalizes the classical Bochner problem to the unit circle, establishing uniqueness of the Lebesgue measure in generating eigenfunction sequences for differential operators.

Findings

Only the Lebesgue measure yields orthogonal Laurent polynomials as eigenfunctions of differential operators.

The result holds even with finitely many exceptions.

The problem is formulated and solved within the framework of CMV matrices.

Abstract

A classical result due to Bochner classifies the orthogonal polynomials on the real line which are common eigenfunctions of a second order linear differential operator. We settle a natural version of the Bochner problem on the unit circle which answers a similar question concerning orthogonal Laurent polynomials and can be formulated as a bispectral problem involving CMV matrices. We solve this CMV bispectral problem in great generality proving that, except the Lebesgue measure, no other one on the unit circle yields a sequence of orthogonal Laurent polynomials which are eigenfunctions of a linear differential operator of arbitrary order. Actually, we prove that this is the case even if such an eigenfunction condition is imposed up to finitely many orthogonal Laurent polynomials.