| Description |
1 online resource (v, 107 pages) : illustrations |
| Series |
Memoirs of the American Mathematical Society, 0065-9266 ; volume 247, number 1174 |
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Memoirs of the American Mathematical Society ; no. 1174.
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| Contents |
Cover; Title page; Chapter 1. Introduction; Chapter 2. Physical motivations; 2.1. Overview; 2.2. Von Neumann measurements; 2.3. Weak values and weak measurements -- the main idea; 2.4. Weak values and weak measurements -- mathematical aspects; 2.5. Large weak values and superoscillations; Chapter 3. Basic mathematical properties of superoscillating sequences; 3.1. Superoscillating sequences; 3.2. Test functions and their Fourier transforms; 3.3. Approximations of functions in (ℝ); Chapter 4. Function spaces of holomorphic functions with growth; 4.1. Analytically Uniform spaces |
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4.2. Convolutors on Analytically Uniform spaces4.3. Dirichlet series; Chapter 5. Schrödinger equation and superoscillations; 5.1. Schrödinger equation for the free particle; 5.2. Approximation by gaussians and persistence of superoscillations; 5.3. Quantum harmonic oscillator; Chapter 6. Superoscillating functions and convolution equations; 6.1. Convolution operators for generalized Schrödinger equations; 6.2. Formal solutions to Cauchy problems for linear constant coefficients differential equations; 6.3. Differential equations of non-Kowalevski type |
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6.4. An application to the harmonic oscillatorChapter 7. Superoscillating functions and operators; 7.1. A quick review on operators; 7.2. Superoscillations and operators; Chapter 8. Superoscillations in (3); 8.1. The weak value of the operator exp( ℒ_{ }\myupn ); 8.2. Asymptotic expansion for the Wigner functions; Bibliography; Index; Back Cover |
| Summary |
"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."--Page v |
| Notes |
"Volume 247, number 1174 (seventh of 7 numbers), May 2017." |
| Bibliography |
Includes bibliographical references (pages 99-105) and index |
| Notes |
Print version record |
| Subject |
Fluctuations (Physics)
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Fourier series.
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Quantum theory.
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Quantum Theory
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Fluctuations (Physics)
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Fourier series.
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Quantum theory.
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| Form |
Electronic book
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| Author |
Colombo, Fabrizio, author.
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Sabadini, Irene, 1965- author.
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Struppa, Daniele Carlo, 1955- author.
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Tollaksen, Jeff, author
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American Mathematical Society, publisher
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| ISBN |
9781470437091 |
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1470437090 |
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