Browsing by Author "Zurrián, Ignacio Nahuel"

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    The algebra of differential operators for a Gegenbauer weight matrix
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2015) Zurrián, Ignacio Nahuel
    In this work we study in detail the algebra of differential operators D(W) associated with a Gegenbauer matrix weight. We prove that two second order operators generate the algebra, indeed D(W) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W) is a finitely-generated torsion-free module over its center, but it is not at and therefore neither projective. After [Tir11], this is the second detailed study of an algebra D(W) and the first one coming from spherical functions and group representation theory.
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    Reflective Prolate-Spheroidal Operators and the KP/KdV Equations
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Casper, W. Riley; Grünbaum, F. A.; Yakimov, Milen; Zurrián, Ignacio Nahuel
    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 KdV equation. We prove a general theorem that the integral operator associated to every wave function in the infinite dimensional Adelic Grassmannian Gr ad of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of 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 90° rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by Airault-McKean-Moser and Krichever, respectively, in the late 70's. Many novel examples are presented.