Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal de Mathématiques Pures et Appliquées

Superoscillating sequences as solutions of generalized Schrödinger equations

Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Daniele Carlo Struppa, Jeff Tollaksen

Published Jul 3, 2014• 48 citations
Publisher
DOI
10.1016/j.matpur.2014.07.001

Related papers

Schrödinger evolution of superoscillations under different potentials

Yakir Aharonov, Fabrizio Colombo, Daniele Carlo Struppa +1 more

Quantum Studies Mathematics and Foundations•Mar 6, 2018•30 citations

Simple solution of the generalized Schrödinger equations

Mario Schönberg

Il Nuovo Cimento•Aug 1, 1954•8 citations

Schrödinger evolution of superoscillations with $$\delta $$- and $$\delta '$$-potentials

Yakir Aharonov, Jussi Behrndt, Fabrizio Colombo +1 more

Abstract In this paper, we study the time persistence of superoscillations as the initial data of the time-dependent Schrödinger equation with $$\delta $$ δ - and $$\delta '$$ δ ′ -potentials. It is shown that the sequence of solutions converges uniformly on compact sets, whenever the initial data converge in the topology of the entire function space $$A_1(\mathbb {C})$$ A 1 ( C ) . Convolution operators acting in this space are our main tool. In particular, a general result about the existence of such operators is proven. Moreover, we provide an explicit formula as well as the large time asymptotics for the time evolution of a plane wave under $$\delta $$ δ - and $$\delta '$$ δ ′ -potentials.

arXiv:1911.04693•Quantum Studies Mathematics and Foundations•Dec 13, 2019•0 citations

The Mathematics of Superoscillations

Yakir Aharonov, Fabrizio Colombo, Irene Sabadini +2 more

In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations . Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.

arXiv:1511.01938•Memoirs of the American Mathematical Society•Feb 1, 2017•86 citations

A unified approach to Schrödinger evolution of superoscillations and supershifts

Yakir Aharonov, Jussi Behrndt, Fabrizio Colombo +1 more

Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time-dependent Schrödinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper, we give a unified approach to determine the supershift property for the solution of the time-dependent one-dimensional Schrödinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green's function, but not its explicit form. With this efficient general technique, we are able to treat various potentials.

arXiv:2102.11795•Journal of Evolution Equations•Mar 1, 2022•0 citations

Generalized Schrödinger representations

Annals of Physics•Jan 1, 1969•0 citations
Schrödinger evolution of superoscillations with δ\delta δ- and δ′\delta 'δ′-potentials
Abstract In this paper, we study the time persistence of superoscillations as the initial data of the time-dependent Schrödinger equation with δ\delta δ δ - and δ′\delta 'δ′ δ ′ -potentials. It is shown that the sequence of solutions converges uniformly on compact sets, whenever the initial data converge in the topology of the entire function space A1(C)A_1(\mathbb {C})A1​(C) A 1 ( C ) . Convolution operators acting in this space are our main tool. In particular, a general result about the existence of such operators is proven. Moreover, we provide an explicit formula as well as the large time asymptotics for the time evolution of a plane wave under δ\delta δ δ - and δ′\delta 'δ′ δ ′ -potentials.
In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations . Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time-dependent Schrödinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper, we give a unified approach to determine the supershift property for the solution of the time-dependent one-dimensional Schrödinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green's function, but not its explicit form. With this efficient general technique, we are able to treat various potentials.