Superoscillating Sequences and Supershifts for Families of Generalized Functions

Abstract

We construct a large class of superoscillating sequences, more generally of $${\mathscr {F}}$$ F -supershifts, where $${\mathscr {F}}$$ F 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 $$\lambda \in {\mathbb {R}}$$ λ ∈ R . The frame in which we introduce such families is that of the evolution through Schrödinger equation $$(i\partial /\partial t - {\mathscr {H}}(x))(\psi )=0$$ ( i ∂ / ∂ t - H ( x ) ) ( ψ ) = 0 ($${\mathscr {H}}(x) = -(\partial ^2/\partial x^2)/2 + V(x)$$ H ( x ) = - ( ∂ 2 / ∂ x 2 ) / 2 + V ( x ) ), V being a suitable potential). If $${\mathscr {F}}= \{(t,x) \mapsto \varphi _\lambda (t,x)\,;\, \lambda \in {\mathbb {R}}\}$$ F = { ( t , x ) ↦ φ λ ( t , x ) ; λ ∈ R } , where $$\varphi _\lambda $$ φ λ is evolved from the initial datum $$x\mapsto e^{i\lambda x}$$ x ↦ e i λ x , $${\mathscr {F}}$$ F -supershifts will be of the form $$\{\sum _{j=0}^N C_j(N,a) \varphi _{1-2j/N}\}_{N\ge 1}$$ { ∑ j = 0 N C j ( N , a ) φ 1 - 2 j / N } N ≥ 1 for $$a\in {\mathbb {R}}{\setminus }[-1,1]$$ a ∈ R \ [ - 1 , 1 ] , taking $$C_j(N,a) =\left( {\begin{array}{c}N\\ j\end{array}}\right) (1+a)^{N-j}(1-a)^j/2^N$$ C j ( N , a ) = N j ( 1 + a ) N - j ( 1 - a ) j / 2 N . 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 $$\mathrm{Exp}({\mathbb {C}})$$ Exp ( C ) . 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 $${\mathscr {F}}$$ F -supershift, $${\mathscr {F}}$$ F being a family of $$C^\infty $$ C ∞ functions or distributions in (t, x), to that where $${\mathscr {F}}$$ F is a family of hyperfunctions in x, depending on t as a parameter.

A proposal to characterize and quantify superoscillations

“…The properties of superoscillations have been the object of extensive study, both from a mathematical [52][53][54][55][56][57][58][59][60][61][62][63] and a physical [64][65][66][67][68][69][70][71][72][73][74][75][76][77] perspective. However, even with all the progress that has been made so far, a clear-cut, rigorous definition of what it means for a function to display superoscillations is still arguably missing.…”

Evolution of superoscillations for spinning particles

“…The literature on superoscillations is quite large, and without claiming completeness some of the most relevant and recent results are contained in the papers [2]- [7], [12], [15], [25], [31] and [32] where the issue of persistence of superoscillatory behavior when evolved under the Schrödinger equation is considered. The papers [18]- [20], [26]- [29] and [35] are mostly concerned with the physical nature of superoscillations, while papers [10], [11], [13]- [14], [21]- [24] develop in depth the mathematical theory of superoscillations. Recently, [8] introduced a new method to generate superoscillating functions.…”

The general theory of superoscillations and supershifts in several variables

“…The literature on superoscillations is quite large, and without claiming completeness we have tried to mention some of the most relevant (and recent) results. Papers [2]- [7], [12], [15], [25], [28] and [29] deal with the issue of permanence of superoscillatory behavior when evolved under a suitable Schrödinger equation; papers [18]- [20], [26]- [27] and [30] are mostly concerned with the physical nature of superoscillations, while papers [10], [11], [13]- [14], [21]- [24] develop in depth the mathematical theory of superoscillations. Finally we have cited [7] as a good reference for the state of the art on the mathematics of superoscillations until 2017, and the Roadmap on Superoscillations [17], where the most recent advances in superoscillations and their applications to technology are well explained by the leading experts in this field.…”