Algebras of Commuting Differential Operators for Kernels of Airy Type

Abstract

Instances of commuting differential and integral operators were discovered by C. Tracy and H. Widom and used to derive asymptotic expansions of Fredholm determinants. Recently, we proved that all rational, symmetric Darboux transformations of the Bessel, Airy, and exponential bispectral functions vastly generalize these examples. In this paper, we give a classification of the Airy family using a differential Galois group action on the Lagrangian locus of the Airy adelic Grassmannian and initiate the study of the full algebra of differential operators commuting with an integral operator. We obtain explicit formulas for the two differential operators of lowest orders that commute with the level one and two integral operators obtained in the Darboux process. Both pairs commute with each other and, in the level one case, are shown to satisfy an algebraic relation defining an elliptic curve.