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# Reflective prolate-spheroidal operators and the KP/KdV equations

Edited by Srinivasa S. R. Varadhan, Courant Institute of Mathematical Sciences, New York, NY, and approved August 5, 2019 (received for review April 9, 2019)

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## Significance

Pairs of commuting integral and differential operators have been constructed on a case-by-case basis in the analysis of various spectral problems in signal processing, random matrix theory, and integrable systems. We present a unified general construction of commuting pairs based on the intrinsic properties of symmetries of soliton equations. A key ingredient in it is a method proving that the integral operators associated to the points of infinite-dimensional families of solutions of soliton equations canonically “reflect” differential operators. This in turn is used to give examples of interesting commuting pairs.

## Abstract

Commuting integral and differential operators connect the topics of signal processing, random matrix theory, and integrable systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the Korteweg–de Vries (KdV) equation. We prove a general theorem that the integral operator associated to every wave function in the infinite-dimensional adelic Grassmannian *Definition 1* below). This intrinsic property is shown to follow from the symmetries of Grassmannians of Kadomtsev–Petviashvili (KP) wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson’s sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A *Commun. Pure Appl. Math.* 30, 95–148 (1977)] and [Krichever, *Funkcional. Anal. i Priložen.* 12, 76–78 (1978)], respectively, in the late 1970s. Many examples are presented.

- prolate-spheroidal integral operators
- reflectivity
- rational solutions of the KdV and KP equations
- Wilson’s adelic Grassmannian

## Footnotes

↵

^{1}W.R.C., F.A.G., M.Y., and I.Z. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: yakimov{at}math.lsu.edu.

Author contributions: W.R.C., F.A.G., M.Y., and I.Z. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.

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