Krall-Jacobi commutative algebras of partial differential operators

TL;DR

This paper constructs a broad class of rotation-invariant commutative algebras of partial differential operators, extending previous one-dimensional theories and confirming a conjecture about Krall-Jacobi algebras.

Contribution

It introduces a new construction method for Krall-Jacobi algebras of PDEs, providing a detailed description and proving a conjecture by Haine.

Findings

Established a new family of commutative PDE algebras invariant under rotations.

Provided a detailed description of the one-dimensional theory of these algebras.

Confirmed Haine's conjecture on the explicit characterization of Krall-Jacobi algebras.

Abstract

We construct a large family of commutative algebras of partial differential operators invariant under rotations. These algebras are isomorphic extensions of the algebras of ordinary differential operators introduced by Grunbaum and Yakimov corresponding to Darboux transformations at one end of the spectrum of the recurrence operator for the Jacobi polynomials. The construction is based on a new proof of their results which leads to a more detailed description of the one-dimensional theory. In particular, our approach establishes a conjecture by Haine concerning the explicit characterization of the Krall-Jacobi algebras of ordinary differential operators.