Gauss sums, superoscillations and the Talbot carpet

TL;DR

This paper explores the connection between Schrödinger equation evolution, Gauss sums, and superoscillations, providing explicit recovery methods and analyzing spectral properties using the Galilean transform.

Contribution

It introduces a novel approach to recover quadratic Gauss sums via Schrödinger evolution and superoscillations, utilizing the Galilean transform for detailed spectral analysis.

Findings

Explicit optical recovery of Gauss sums from Schrödinger evolution

Asymptotic recovery of Gauss sums using superoscillations

Detailed understanding of exponential evolution in periodic potentials

Abstract

We consider the evolution, for a time-dependent Schr\"odinger equation, of the so called Dirac comb. We show how this evolution allows us to recover explicitly (indeed optically) the values of the quadratic generalized Gauss sums. Moreover we use the phenomenon of superoscillatory sequences to prove that such Gauss sums can be asymptotically recovered from the values of the spectrum of any sufficiently regular function compactly supported on $\R$. The fundamental tool we use is the so called Galilean transform that was introduced and studied in the context on non-linear time dependent Schr\"odinger equations. Furthermore, we utilize this tool to understand in detail the evolution of an exponential $e^{i\omega x}$ in the case of a Schr\"odinger equation with time-independent periodic potential.