Superoscillating Sequences and Supershifts for Families of Generalized Functions
Superoscillating Sequences and Supershifts for Families of Generalized Functions
复制标题
广义函数族的超振荡序列和超移
DOI：
--
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发表时间：
2019
影响因子：
0.8
通讯作者：
A. Yger
中科院分区：
文献类型：
--
作者：
F. Colombo;I. Sabadini;D. Struppa;A. Yger
我们构造一大类超振荡序列，更一般的是 F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document}-supershifts，其中 F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document} 是一系列 (t, x) 中的平滑函数（分别是 (t, x) 中的分布，或 x 中的超函数，具体取决于参数 t），索引为 λεR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{文档}$$\lambda \in {\mathbb {R}}$$\end{文档}。我们引入这些族的框架是通过薛定谔方程 (i∂/∂t-H(x))(ψ)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} 演化的框架\setlength{\oddsidemargin}{-69pt} \begin{文档}$$(i\partial /\partial t - {\mathscr {H}}(x))(\psi )=0$$\end{文档} (H(x)=-(∂2/∂x2)/2+V(x)\documentclass[12pt]{最小} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{文档}$${\mathscr {H}}(x) = -(\partial ^2/\partial x^2)/2 + V(x)$$\end{document})，V 是合适的势)。如果 F={(t,x)↦φλ(t,x);λ∈R}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{文档}$${\mathscr {F}}= \{(t,x) \mapsto \varphi _\lambda (t,x)\,;\, \lambda \in {\mathbb {R}}\}$$\end{文档}，其中 φλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _\lambda $$\end{document} 由初始数据 x↦eiλx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} 演变而来\usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\mapsto e^{i\lambda x}$$\end{document}, F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document}-supershifts 将采用以下形式{Σj=0NCj(N,a)φ1-2j/N}N≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\sum _{j=0}^N C_j(N,a) \varphi _{1-2j/N}\}_{N\ge 1}$$\end{document} for a∈R\[-1,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in {\mathbb {R}}{\setminus }[-1,1]$$\end{document}，取Cj(N,a)=Nj(1+a)N-j(1-a)j/2N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{文档}$$C_j(N,a) =\left( {\begin{array}{c}N\\ j\end{array}}\right) (1+a)^{N-j}(1-a)^j/2^N$$\end{文档}.我们的结果依赖于这样一个事实：菲涅耳类型的积分算子控制着薛定谔方程的演化，就像在光学衍射中一样，这些算子连续作用于整个函数 Exp(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} 的加权代数\usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{文档}$$\mathrm{Exp}({\mathbb {C}})$$\end{文档}。特别分析量子谐振子情况迫使我们，为了考虑菲涅耳算子中的静止相位消失时发生的演化数据的奇点，扩大 F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} 的概念\usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document}-supershift, F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document} 是 C∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} 家族\usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document} 函数或 (t, x) 中的分布，其中 F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document} 是 x 中的一系列超函数，依赖于 t 作为参数。
We construct a large class of superoscillating sequences, more generally of F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document}-supershifts, where F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document} is a family of smooth functions in (t, x) (resp. distributions in (t, x), or hyperfunctions in x depending on the parameter t) indexed by λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in {\mathbb {R}}$$\end{document}. The frame in which we introduce such families is that of the evolution through Schrödinger equation (i∂/∂t-H(x))(ψ)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(i\partial /\partial t - {\mathscr {H}}(x))(\psi )=0$$\end{document} (H(x)=-(∂2/∂x2)/2+V(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {H}}(x) = -(\partial ^2/\partial x^2)/2 + V(x)$$\end{document}), V being a suitable potential). If F={(t,x)↦φλ(t,x);λ∈R}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}= \{(t,x) \mapsto \varphi _\lambda (t,x)\,;\, \lambda \in {\mathbb {R}}\}$$\end{document}, where φλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _\lambda $$\end{document} is evolved from the initial datum x↦eiλx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\mapsto e^{i\lambda x}$$\end{document}, F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document}-supershifts will be of the form {∑j=0NCj(N,a)φ1-2j/N}N≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\sum _{j=0}^N C_j(N,a) \varphi _{1-2j/N}\}_{N\ge 1}$$\end{document} for a∈R\[-1,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in {\mathbb {R}}{\setminus }[-1,1]$$\end{document}, taking Cj(N,a)=Nj(1+a)N-j(1-a)j/2N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_j(N,a) =\left( {\begin{array}{c}N\\ j\end{array}}\right) (1+a)^{N-j}(1-a)^j/2^N$$\end{document}. Our results rely on the fact that integral operators of the Fresnel type govern, as in optical diffraction, the evolution through the Schrödinger equation, such operators acting continuously on the weighted algebra of entire functions Exp(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Exp}({\mathbb {C}})$$\end{document}. Analyzing in particular the quantum harmonic oscillator case forces us, in order to take into account singularities of the evolved datum that occur when the stationary phasis in the Fresnel operator vanishes, to enlarge the notion of F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document}-supershift, F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document} being a family of C∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document} functions or distributions in (t, x), to that where F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document} is a family of hyperfunctions in x, depending on t as a parameter.