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 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 Grad gives rise to an integral operator TW, acting on L2(Γ) for a contour Γ⊂C, which reflects a differential operator R(z,∂z) in the sense that R(−z,−∂z)∘TW=TW∘R(w,∂w) on a dense subset of L2(Γ). 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 ψW(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 ψW(x,−z) 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 ψW(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 1970sFil: Casper, W. Riley. State University of Louisiana; Estados UnidosFil: Grünbaum, Francisco Alberto. University of California at Berkeley; Estados UnidosFil: Yakimov, Milen. State University of Louisiana; Estados UnidosFil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin

Algebras of commuting differential operators for integral kernels of Airy type

Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix theory. Very recently, it has been proved that all rational, symmetric Darboux transformations of the Bessel, Airy, and exponential bispectral functions give rise to commuting integral and differential operators [6, 7, 8], vastly generalizing the known examples in the literature. In this paper, we give a classification of the the rational symmetric Darboux transformations of the Airy function in terms of the fixed point submanifold of a differential Galois group acting on the Lagrangian locus of the (infinite dimensional) Airy Adelic Grassmannian and initiate the study of the full algebra of differential operators commuting with each of the integral operators in question. We leverage the general theory of [8] to obtain explicit formulas for the two differential operators of lowest orders that commute with each of the level one and two integral operators obtained in the Darboux process. Moreover, we prove that each pair of differential operators commute with each other. The commuting operators in the level one case are shown to satisfy an algebraic relation defining an elliptic curve.Fil: Casper, W. Riley. California State University, Fullerton; Estados UnidosFil: Grünbaum, Francisco Alberto. University of California at Berkeley; Estados UnidosFil: Yakimov, Milen. Northeastern University; Estados UnidosFil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin

A Sato-Krichever Theory for Fractional Differential Operators

Fractional differential (and difference) operators play a role in a number of diverse settings: integrable systems, mirror symmetry, Hurwitz numbers, the Bethe ansatz equations. We prove extensions of the three major results on algebras of commuting (ordinary) differentials operators to the setting of fractional differential operators: (1) the Burchnall-Chaundy theorem that a pair of commuting differential operators is algebraically dependent, (2) the classification of maximal commutative algebras of differential operators in terms of Sato's theory and (3) the Krichever correspondence constructing those of rank 1 in an algebro-geometric way. Unlike the available proofs of the Burchnall-Chaundy theorem which use the action of one differential operator on the kernel of the other, our extension to the fractional case uses bounds on orders of fractional differential operators and growth of algebras, which also presents a new and much shorter proof of the original result. The second main theorem is achieved by developing a new tool of the spectral field of a point in Sato's Grassmannian, which carries more information than the widely used notion of spectral curve of a KP solution. Our Krichever type correspondence for fractional differential operators is based on infinite jet bundles.Comment: 32 page