论文标题
反射性倾斜的球员和Adelic Grassmannian
Reflective prolate-spheroidal operators and the adelic Grassmannian
论文作者
论文摘要
从1960年代在限制时间段内的Landau，Pollak和Slepian的工作开始，通勤对成对的积分和差异操作员在信号处理，随机矩阵理论和可集成的系统中发挥了关键作用。以前，这种对是通过临时方法构建的，该方法起作用是因为可以通过直接计算找到低阶的通勤操作员。我们描述了一种解决这些问题的一般方法，证明威尔逊的无限尺寸$ w $ w $ a $ \ mathrm gr^ad $都会引起整体操作员$ t_w $，作用于$ l^2（γ2（γ）$，用于轮廓$γ\ subset \ subset \ subbb c $ $ $ $ $ $ $ $ $ $（z），这是一定的（z $ $）。 $ r（-z， - \ partial_z）\ circ t_w = t_w \ circ r（w，\ partial_w）$在$ l^2（γ）$的密集子集上。通过使用来自集成系统的分析方法和方法，我们表明可以从相关的双光谱函数的傅立叶代数$ψ_W（x，z）$构建反射的差分运算符。该代数相对于分叉的大小又使用代数几何方法确定。 Adelic Grassmannian的四个相关性的内在特性自然会导致我们考虑反射性能代替普通的交换性。此外，我们证明了广义拉普拉斯的时间频段有限运算符会通过所有等级给出的内核转换，一个双光谱函数$ψ_W（x，-z）$反映了一个差异操作员。 $ 90^\ circ $旋转参数用于证明具有内核$ψ_W（x，iz）$的广义傅立叶变换的时间频段有限运算符承认通勤差分运算符。这些方法产生了具有巨大的积分运算符的集合，具有岩体 - 球状特性，与Krichever在1970年代后期引入了KP层次结构消失的所有合理解决方案的波函数。
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 $\mathrm Gr^ad$ gives rise to an integral operator $T_W$, acting on $L^2(Γ)$ for a contour $Γ\subset\mathbb C$, which reflects a differential operator $R(z,\partial_z)$ in the sense that $R(-z,-\partial_z)\circ T_W=T_W\circ R(w,\partial_w)$ on a dense subset of $L^2(Γ)$. 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 $ψ_W(x,z)$. 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 $ψ_W(x,-z)$ reflect a differential operator. A $90^\circ$ rotation argument is used to prove that the time-band limited operators of the generalized Fourier transforms with kernels $ψ_W(x,iz)$ 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 1970s.
