Artículo
Algebras of commuting differential operators for integral kernels of Airy type
Fecha de publicación:
12/2021
Editorial:
Cornell University
Revista:
arXiv
ISSN:
2331-8422
Idioma:
Inglés
Tipo de recurso:
Artículo publicado
Clasificación temática:
Resumen
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.
Palabras clave:
Mathematical physics
,
Rings and algebras
,
Spectral theory
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Articulos(CIEM)
Articulos de CENT.INV.Y ESTUDIOS DE MATEMATICA DE CORDOBA(P)
Articulos de CENT.INV.Y ESTUDIOS DE MATEMATICA DE CORDOBA(P)
Citación
Casper, W. Riley; Grünbaum, Francisco Alberto; Yakimov, Milen; Zurrián, Ignacio Nahuel; Algebras of commuting differential operators for integral kernels of Airy type; Cornell University; arXiv; 12-2021; 1-19
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