AbstractIn 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.