Algebras of commuting differential operators for kernels of Airy type. (English) Zbl 07827905
Basor, Estelle (ed.) et al., Toeplitz operators and random matrices. In memory of Harold Widom. Cham: Birkhäuser. Oper. Theory: Adv. Appl. 289, 229-256 (2022).
Summary: 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.
For the entire collection see [Zbl 1519.47002].
For the entire collection see [Zbl 1519.47002].
MSC:
| 47G10 | Integral operators |
| 16S32 | Rings of differential operators (associative algebraic aspects) |
| 12H05 | Differential algebra |
| 14H70 | Relationships between algebraic curves and integrable systems |
Keywords:
commuting integral and differential operators; bispectral functions; Fourier algebras; adelic Grassmannian; differential Galois groupsReferences:
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