Evolution of Superoscillations in the Dirac Field
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Evolution of Superoscillations in the Dirac Field Fabrizio Colombo1 · Giovanni Valente1 Received: 29 January 2020 / Accepted: 14 September 2020 © The Author(s) 2020
Abstract Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The study of the evolution of superoscillations as initial datum of field equations requires the notion of supershift, which generalizes the concept of superoscillations. The present paper has a dual purpose. The first one is to give an updated and self-contained explanation of the strategy to study the evolution of superoscillations by referring to the quantum-mechanical Schrödinger equation and its variations. The second purpose is to treat the Dirac equation in relativistic quantum theory. The treatment of the evolution of superoscillations for the Dirac equation can be deduced by recent results on the Klein–Gordon equation, but further additional considerations are in order, which are fully described in this paper. Keywords Superoscillating functions · Convolution operators · Schrödinger equation · Dirac equation · Entire functions with growth conditions Mathematics Subject Classification 32A15 · 32A10 · 47B38
1 Introduction to Superoscillations and the Supershift Property Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in weak values in quantum mechanics (see [3, 12, 14, 26]) and in several other fields of science and technology, as it will be mentioned in the sequel. A natural problem, suggested by Aharonov, is to study the evolution of weak-values-superoscillations as initial datum of Schrödinger equation or as initial condition of other quantum field equations such are Klein–Gordon, see [10], or Dirac equation which is treated in this paper. * Giovanni Valente [email protected] Fabrizio Colombo [email protected] 1
Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, 20133 Milan, Italy
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Foundations of Physics
In order to explain the topic and our main results we consider in this section some heuristic facts and properties of superoscillations and we introduce a diagram to visualize them. Furthermore, we discuss, from an intuitive point of view, the notion of supershift property of the solutions of differential equations which generalizes the notion of superoscillations. The supershift property is the crucial concept to investigate the evolution of superoscillatory initial datum by all field equations. In the last decade a systematic study of superoscillating functions has been carried out also from the mathematical point of view. In fact, the rigorous treatment of the evolution of superoscillations, as initial datum of the Schrödinger equation, required some sophisticated mathematical tools, as it has been shown for example in [1, 2, 4–8, 16, 17, 27–29] and partially summarized in [9]. The prototypical function arising in Aharonov’s weak values i
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