Integral operators, bispectrality and growth of Fourier algebras

DOI10.1515/CRELLE-2019-0031arXiv1807.09314WikidataQ127184870 ScholiaQ127184870MaRDI QIDQ6304600

Publication date: 24 July 2018

Abstract: In the mid 80's it was conjectured that every bispectral meromorphic function

p s i (x, y)

gives rise to an integral operator

K_{p s i} (x, y)

which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions

p s i (x, y)

where the commuting differential operator is of order

l e q 6

. We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second and first order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel

K_{p s i} (x, y)

and a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Connections of hypergeometric functions with groups and algebras, and related topics (33C80) Integral operators (47G10) Relationships between algebraic curves and integrable systems (14H70) General integral transforms (44A05)

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