Document Type
Article
Publication Date
3-18-2022
Abstract
We construct a large class of superoscillating sequences, more generally of F-supershifts, where 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 λ ∈ R. The frame in which we introduce such families is that of the evolution through Schrödinger equation (i∂/∂t−H (x))(ψ) = 0 (H (x) = −(∂2/∂x2)/2+V (x)), V being a suitable potential). If F = {(t, x) → ϕλ(t, x) ; λ ∈ R}, where ϕλ is evolved from the initial datum x → eiλx , F-supershifts will be of the form {Nj =0 Cj (N, a)ϕ1−2 j/N }N≥1 for a ∈ R\[−1, 1], taking Cj (N, a) = Nj (1 + a)N−j (1 − a) j /2N . 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). 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-supershift, F being a family of C∞ functions or distributions in (t, x), to that where F is a family of hyperfunctions in x, depending on t as a parameter.
Recommended Citation
Colombo, F., Sabadini, I., Struppa, D.C. et al. Superoscillating Sequences and Supershifts for Families of Generalized Functions. Complex Anal. Oper. Theory 16, 34 (2022). https://doi.org/10.1007/s11785-022-01211-0
Peer Reviewed
1
Copyright
The authors
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
Comments
This article was originally published in Complex Analysis and Operator Theory, volume 16, in 2022. https://doi.org/10.1007/s11785-022-01211-0