Superoscillating sequences and supershifts for families of generalized functions

TL;DR

This paper constructs and analyzes a broad class of superoscillating sequences and supershifts generated via Schrödinger evolution, extending the concept to hyperfunctions to handle singularities, and demonstrates their convergence properties.

Contribution

It introduces a new framework for superoscillating sequences and supershifts using Schrödinger evolution, including hyperfunctions, and proves their convergence.

Findings

Constructed a large class of superoscillating sequences and supershifts.

Proved locally uniform convergence of derivatives of supershifts.

Extended the notion of supershifts to hyperfunctions for the quantum harmonic oscillator.

Abstract

We construct in this paper a large class of superoscillating sequences, more generally of $\mathscr F$-supershifts, where $\mathscr F$ is a family of smooth functions (resp. distributions, hyperfunctions) indexed by a real parameter $\lambda\in \R$. The key model we introduce in order to generate such families is the evolution through a Schr\"odinger equation $(i\partial/\partial t - \mathscr H(x))(\psi)=0$ with a suitable hamiltonian $\mathscr H$, in particular a suitable potential $V$ when $\mathscr H(x) = -(\partial^2/\partial x^2)/2 + V(x)$. The family $\mathscr F$ is in this case $\mathscr F= \{(t,x) \mapsto \varphi_\lambda(t,x)\,;\, \lambda \in \R\}$, where $\varphi_\lambda$ is evolved from the initial datum $x\mapsto e^{i\lambda x}$. Then $\mathscr F$-supershifts will be of the form $\{\sum_{j=0}^N C_j(N,a) \varphi_{1-2j/N}\}_{N\geq 1}$ for $a\in \R\setminus [-1,1]$, taking $C_j(N,a) =\binom{N}{j}(1+a)^{N-j}(1-a)^j/2^N$. We prove the locally uniform convergence of derivatives of the supershift towards corresponding derivatives of its limit. We analyse in particular the case of the quantum harmonic oscillator, which forces us, in order to take into account singularities of the evolved datum, to enlarge the notion of supershifts for families of functions to a similar notion for families of hyperfunctions, thus beyond the frame of distributions.