Persistence of superoscillations under the Schrödinger equation

Elodie Pozzi; Brett D. Wick

doi:10.3934/eect.2021029

Article Contents

2022, Volume 11, Issue 3: 869-894. Doi: 10.3934/eect.2021029

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Persistence of superoscillations under the Schrödinger equation

Elodie Pozzi^1, , and
Brett D. Wick^2,

1.
Department of Mathematics & Statistics, Saint Louis University, Saint Louis, MO 63103, USA
2.
Department of Mathematics & Statistics, Washington University in Saint Louis, Saint Louis, MO 63130, USA

^* Corresponding author: Elodie Pozzi
^* Corresponding author: Elodie Pozzi

Received: November 30, 2020

Revised: March 31, 2021

Published: June 2022

The second author is supported in part by a National Science Foundation DMS grant # 1800057 and Australian Research Council – DP 190100970

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Abstract

The goal of this paper is to provide new proofs of the persistence of superoscillations under the Schrödinger equation. Superoscillations were first put forward by Aharonov and have since received much study because of connections to physics, engineering, signal processing and information theory. An interesting mathematical question is to understand the time evolution of superoscillations under certain Schrödinger equations arising in physics. This paper provides an alternative proof of the persistence of superoscillations by some elementary convergence facts for sequence and series and some connections with certain polynomials and identities in combinatorics. The approach given opens new perspectives to establish persistence of superoscillations for the Schrödinger equation with more general potentials.

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Mathematics Subject Classification: Primary: 35Q41, 81Q05; Secondary: 35C10, 47F05.

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