Reflective prolate-spheroidal operators and the adelic Grassmannian

Beginning with the work of Landau, Pollak and Slepian in the 1960s on time-band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point W of Wilson's infinite dimensional adelic Grassmannian ${\mathrm{Gr}}^{\mathrm{ad}}$ gives rise to an integral operator $T_W$, acting on $L^2\left(\Gamma \right)$ for a contour $\Gamma \subset C$, which reflects a differential operator with rational coefficients $R(z,{\partial }_z)$ in the sense that $R(-z,-{\partial }_z)\circ T_W=T_W\circ R(w,{\partial }_w)$ on a dense subset of $L^2\left(\Gamma \right)$. By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function ${\psi }_W(x,z)$. The exact size of this algebra with respect to a bifiltration is in turn determined using algebro-geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time-band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions ${\psi }_W(x,-z)$ always reflect a differential operator. A 90° rotation argument is used to prove that the time-band limited operators of the generalized Fourier transforms with kernels ${\psi }_W(x,iz)$ admit a commuting differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.