Integral operators, bispectrality and growth of Fourier algebras. (English) Zbl 1540.47069
Summary: In the mid 1980s it was conjectured that every bispectral meromorphic function \(\psi(x,y)\) gives rise to an integral operator \(K_{\psi}(x,y)\) which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions \(\psi(x,y)\) where the commuting differential operator is of order \(\leq 6\). 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 \(K_{\psi}(x,y)\) leading to 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.
MSC:
| 47G10 | Integral operators |
| 44A05 | General integral transforms |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
| 14H70 | Relationships between algebraic curves and integrable systems |
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