 
Summary: THE PROLATE SPHEROIDAL PHENOMENON AND
BISPECTRAL ALGEBRAS OF ORDINARY DIFFERENTIAL
OPERATORS
F. ALBERTO GRšUNBAUM AND MILEN YAKIMOV
Abstract. Landau, Pollak, Slepian, and Tracy, Widom discovered that cer
tain integral operators with so called Bessel and Airy kernels possess commut
ing differential operators and found important applications of this phenomenon
in timeband limiting and random matrix theory. In this paper we announce
that very large classes of integral operators derived from bispectral algebras
of rank 1 and 2 (parametrized by lagrangian grassmannians of infinitely large
size) have this property. The above examples come from special points in these
grassmannians.
1. Introduction
It was discovered by Landau, Pollak, Slepian [23, 18, 19, 22] and Tracy, Widom
[25, 26] that certain integral operators associated to the Airy and Bessel special
functions posses commuting differential operators. They found important applica
tions of this to timeband limiting, and to the study of asymptotics of Fredholm
determinants, relevant to scaling limits of random matrix models. We call this
phenomenon the prolate spheroidal phenomenon.
On the other hand, the problem of bispectrality was posed [8] about 20 years
