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We consider the evolution, for a time-dependent Schrödinger 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 Schrödinger equations. Furthermore, we utilize this tool to understand in detail the evolution of an exponential ei!x in the case of a Schrödinger equation with time-independent periodic potential.


NOTICE: this is the author’s version of a work that was accepted for publication in Journal de Mathématiques Pures et Appliquées. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal de Mathématiques Pures et Appliquées, volume 147, in 2021.

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