Integral operators, bispectrality and growth of Fourier algebras

In the mid 80's 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 $K_{\psi}(x,y)$ 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.