Hankel operators that commute with second-order differential operators

Abstract

Suppose that $\Gamma$ is a continuous and self-adjoint Hankel operator on $L^2(0, \infty)$ and that $Lf=-(d/dx(a(x)df/dx))+b(x)f(x)$ with $a(0)=0$. If $a$ and $b$ are both quadratic, hyperbolic or trigonometric functions, and $\phi$ satisfies a suitable form of Gauss's hypergeometric equation, or the confluent hypergeometric equation, then $L\Gamma =\Gamma L$. The paper catalogues the commuting pairs $\Gamma$ and $L$, including important cases in random matrix theory. There are also results proving rapid decay of the singular numbers of Hankel integral operators with kernels that are analytic and of exponential decay in the right half plane.Comment: 18 page

Decay of Hankel singular values of analytic control systems

“…For discrete-time infinite-dimensional systems this is treated in [3]. For applications in random matrix theory the decay rate of Hankel singular values (under for control theory applications unreasonably strict assumptions) is studied in [2]. For finite-dimensional systems an investi gation of the decay of Hankel singular values was carried out in [1], where also the decay of the eigenvalues of the systems gramians is studied.…”

Kernels and point processes associated with Whittaker functions

“…The column vector Y (z) = √ zW (z) satisfies z(d/dz)Y (z) = Ω(z)Y (z), which resembles the differential equation for generalized Laguerre functions on[33, page 60]. In Remark 5.2 of[2] we obtained a factorization theorem for certain Whittaker kernels which expressly excluded the case of generalized Laguerre functions.…”

“…The function W 0,im also occurs in the spectral decomposition of the Laplace operator over the fundamental domain that arises from the action of SL(2, Z) on the upper half plane; see [21, p 318] for a discussion of Maass cusp forms. In [2], we considered the Hankel operators that commute with second order differential operators, and found the case L 0 and R 0 as Q(vii).…”

“…gives a determinantal point process on [0, ∞) such that random points λ j , ordered by (ii) Another case of Q(vii) from [2] is L −κ commuting with the Hankel operator with scattering function x −1 W κ,1/2 (x) on C ∞ c (0, ∞). If a = 0, then R a is not a Hankel operator but an operator of Howland's type as in [17], and L a does not commute with R −a .…”