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.
Highlights
Introduction to Superoscillations and the SupershiftPropertySuperoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component
Superoscillations are a particular case of supershift, which is the property to be proven when we study the evolution of superoscillations as initial datum of given field equation
We summarize the main result on the supershift property for the Dirac equation, but we postpone the proofs to Sect. 4 of the paper
Summary
Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. In order to understand the meaning of the evolution of superoscillations as initial condition of a given quantum field equation we need the following interpretation of the generalized Fourier sequence Fn(x, a). Since for the particular case of the free particle the solution of Schrödinger equation is written in terms of the exponential function we still speak of superoscillations, but in the general case of nontrivial potentials the functions ( , t, x) are not necessarily the exponential function In this case there arises the natural notion of supershift of the solution where we compute the function ↦ ( , t, x) in infinitely many points 1 − 2j∕n that belong to the interval [−1, 1] , and we determine the value of ( , t, x) in the point λ = a >> 1.
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