Evolution of superoscillatory initial data in several variables in uniform electric field. (30th March 2017)

• Record Type: Journal Article
• Authors: Aharonov, Y; Colombo, F; Sabadini, I; Struppa, D C; Tollaksen, J
• Is Part Of: Journal of physics. Volume 50:Number 18(2017)
• Publication Date: 2017-03-30
• Subjects: superoscillating functions in several variables -- Schrödinger equation in uniform electric field -- Fourier transform; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/aa66d9
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: A superoscillating function is defined by the property that it oscillates faster than its fastest Fourier components. This is mathematically possible because the coefficients of the linear combinations of the band limited components depend on the number of components. This phenomenon was discovered in the context of quantum physics, but it has important applications in a variety of areas, including metrology, antenna theory, and a new theory of superresolution in optics. In this paper we study the evolution of superoscillatory functions in uniform electric field by the Schrödinger equation where we assume that the Hamiltonian contains a even polynomial of the linear momentum p . This includes the classical case but also relativistic corrections of any order. Moreover, we extend our results to the case of several variables using the theory of superoscillating functions in several variables. We conclude by discussing a comparison of our work with the existing literature.