Aharonov-Berry superoscillations are band-limited functions that oscillate faster than their fastest Fourier component. Superoscillations appear in several fields of science and technology, such as Aharonov’s weak measurement in quantum mechanics, in optics, and in signal processing. An important issue is the study of the evolution of superoscillations using the Schrödinger equation when the initial datum is a weak value. Some superoscillatory functions are not square integrable, but they are real analytic functions that can be extended to entire holomorphic functions. This fact leads to the study of the continuity of a class of convolution operators acting on suitable spaces of entire functions with growth conditions. In this paper, we study the evolution of a superoscillatory initial datum in a uniform magnetic field. Moreover, we collect some results on convolution operators that appear in the theory of superoscillatory functions using a direct approach that allows the convolution operators to have non-constant coefficients of polynomial type.
Colombo, F., Gantner, J., and Struppa, D.C., “Evolution of superoscillations for Schrödinger equation in a uniform magnetic field,” J. Math. Phys. 58, 092103 (2017). doi: 10.1063/1.4991489