Integral operators, bispectrality and growth of Fourier algebras

In the mid 1980s it was conjectured that every bispectral meromorphic function ψ ⁢ ( x , y ) {\psi(x,y)} gives rise to an integral operator K ψ ⁢ ( x , y ) {K_{\psi}(x,y)} which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions ψ ⁢ ( x , y ) {\psi(x,y)} where the commuting differential operator is of order ≤ 6 {\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 ψ ⁢ ( x , y ) {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.