Superoscillating Sequences and Supershifts for Families of Generalized Functions

Abstract

AbstractWe 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.