Document Type


Publication Date



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.


This article was originally published in Complex Analysis and Operator Theory, volume 16, in 2022.

Peer Reviewed



The authors

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.