| Abstract: |
| In this talk, we investigate the points of constancy in the piecewise constant solution profiles of the periodic linear Schr$\mathrm{\ddot{o}}$dinger equation with step-function initial data at rational times. We characterize all points of constancy of the solution $u$ and the square of the modulus $|u|^2$ of $u$, respectively. We employ number theoretic techniques, including quadratic Gauss sums and half-Gauss sums. These results establish an intriguing connections between the revival phenomena of dispersive evolution equations on a periodic domain and the classical number theory. |
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