Algebras of commuting differential operators for integral kernels of Airy type

TL;DR

This paper classifies rational Darboux transformations of the Airy function, studies the algebra of differential operators commuting with associated integral operators, and provides explicit formulas and algebraic relations for these operators.

Contribution

It introduces a classification of Darboux transformations for the Airy function and explores the structure of the algebra of commuting differential operators, including explicit formulas and algebraic relations.

Findings

Explicit formulas for the lowest order commuting differential operators.

Proof that pairs of these operators commute with each other.

Commutation relations define an elliptic curve in the level one case.

Abstract

Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix theory. Very recently, it has been proved that all rational, symmetric Darboux transformations of the Bessel, Airy, and exponential bispectral functions give rise to commuting integral and differential operators [6, 7, 8], vastly generalizing the known examples in the literature. In this paper, we give a classification of the the rational symmetric Darboux transformations of the Airy function in terms of the fixed point submanifold of a differential Galois group acting on the Lagrangian locus of the (infinite dimensional) Airy Adelic Grassmannian and initiate the study of the full algebra of differential operators commuting with each of the integral operators in question. We leverage the general theory of [8] to obtain explicit formulas for the two differential operators of lowest orders that commute with each of the level one and two integral operators obtained in the Darboux process. Moreover, we prove that each pair of differential operators commute with each other. The commuting operators in the level one case are shown to satisfy an algebraic relation defining an elliptic curve.