Krall–Jacobi commutative algebras of partial differential operators

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 Grünbaum 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.