Reflective prolate-spheroidal operators and the adelic Grassmannian

TL;DR

This paper develops a unified framework linking adelic Grassmannian points to integral operators with prolate-spheroidal properties, revealing new connections between integrable systems, bispectral functions, and differential operators.

Contribution

It introduces a general approach to construct integral operators from adelic Grassmannian points that reflect differential operators, expanding the theory of time-band limiting and bispectrality.

Findings

Constructed integral operators from adelic Grassmannian points reflecting differential operators.

Linked time-band limited operators to bispectral functions with differential reflection properties.

Applied algebraic and geometric methods to analyze the size and properties of associated algebras.

Abstract

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 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 $\mathrm Gr^ad$ gives rise to an integral operator $T_W$, acting on $L^2(\Gamma)$ for a contour $\Gamma\subset\mathbb C$, which reflects a differential operator $R(z,\partial_z)$ in the sense that $R(-z,-\partial_z)\circ T_W=T_W\circ R(w,\partial_w)$ on a dense subset of $L^2(\Gamma)$. 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 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 in place of plain commutativity. Furthermore, we prove that the time-band limited operators of the generalized Laplace transforms with kernels given by all rank one bispectral functions $\psi_W(x,-z)$ reflect a differential operator. A $90^\circ$ 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.