论文标题
高斯总和，超激动和塔尔伯特地毯
Gauss sums, superoscillations and the Talbot carpet
论文作者
论文摘要
我们考虑所谓的Dirac Comb的时间依赖性Schrödinger方程的演变。我们展示了这种演变如何使我们能够明确（确实是光学）恢复二次概括的高斯总和的值。此外，我们使用升级序列的现象来证明可以从$ \ r $ $ \ r $上紧凑的任何足够规则函数的频谱值中渐近地恢复这样的高斯总和。我们使用的基本工具是在非线性时间依赖性Schrödinger方程的上下文中引入和研究的所谓的Galilean变换。此外，在Schrödinger方程中，我们利用此工具详细了解指数$ e^{iΩx} $的演变。
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 $e^{iωx}$ in the case of a Schrödinger equation with time-independent periodic potential.
