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arXiv:quant-ph/0405065 · Journal of Mathematical Physics

Analysis of superoscillatory wave functions

Matt S. Calder, Achim Kempf

Published Dec 22, 2004• 26 citations
arXivHTMLPDF

Abstract

Surprisingly, differentiable functions are able to oscillate arbitrarily faster than their highest Fourier component would suggest. The phenomenon is called superoscillation. Recently, a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter-intuitive physical effects. Following up on this work, we here present more general methods which allow the calculation of superoscillatory wave functions with custom-designed physical properties. We give concrete examples and we prove results about the limits to superoscillatory behavior. We also give a simple and intuitive new explanation for the exponential computational cost of superoscillations.

DOI
10.1063/1.1825076

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Surprisingly, differentiable functions are able to oscillate arbitrarily faster than their highest Fourier component would suggest. The phenomenon is called superoscillation. Recently, a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter-intuitive physical effects. Following up on this work, we here present more general methods which allow the calculation of superoscillatory wave functions with custom-designed physical properties. We give concrete examples and we prove results about the limits to superoscillatory behavior. We also give a simple and intuitive new explanation for the exponential computational cost of superoscillations.
Abstract. Surprisingly, differentiable functions are able to oscillate arbitrarily faster than their highest Fourier component would suggest. The phenomenon is called superoscillation. Recently, a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter-intuitive physical effects. Following up on this work, we here present more general methods which allow the calculation of superoscillatory wave functions with custom-designed physical properties. We give concrete examples and we prove results about the limits to superoscillatory behavior. We also give a simple and intuitive new explanation for the exponential computational cost of superoscillations. 1 2 1.
Abstract Superoscillatory wave forms, i.e., waves that locally oscillate faster than their highest Fourier component, possess unusual properties that make them of great interest from quantum mechanics to signal processing. However, the more pronounced the desired superoscillatory behavior is to be, the more difficult it becomes to produce, or even only calculate, such highly fine-tuned wave forms in practice. Here, we investigate how this sensitivity to preparation errors scales for a method for constructing superoscillatory functions which is optimal in the sense that it minimizes the energetic expense. We thereby also arrive at very accurate approximations of functions which are so highly superoscillatory that they cannot be calculated numerically. We then investigate to what extent the scaling and sensitivity results for superoscillatory functions on the real line extend to the experimentally important case of superoscillatory functions that are periodic.
Abstract Superoscillating functions, i.e. functions that locally oscillate at a rate faster than their highest Fourier component, are of interest for applications from fundamental physics to engineering. Here, we develop a new method which allows one to construct superoscillations of arbitrarily high frequency and arbitrarily long duration in a computationally efficient way. We also present a method for constructing non-singular Schrödinger potentials whose ground state is a superoscillating wave function.
It has been found that functions can oscillate locally much faster than their Fourier transform would suggest is possible - a phenomenon called superoscillation. Here, we consider the case of superoscillating wave functions in quantum mechanics. We find that they possess rather unusual properties which raise measurement theoretic, thermodynamic and information theoretic issues. We explicitly determine the wave functions with the most pronounced superoscillations, together with their scaling behavior. We also address the question how superoscillating wave functions could be produced.
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.
It has been found that functions can oscillate locally much faster than their Fourier transform would suggest is possible - a phenomenon called superoscillation. Here, we consider the case of superoscillating wave functions in quantum mechanics. We find that they possess rather unusual properties which raise measurement theoretic, thermodynamic and information theoretic issues. We explicitly determine the wave functions with the most pronounced superoscillations, together with their scaling behavior. We also address the question how superoscillating wave functions could be produced.