AI 中文总结
本文证明了有界变差且含至少一个跳跃的粗糙数据对应的环面自由薛定谔演化密度几乎处处为分形，其临界索伯列夫质量的发散速率可探测色散关系，相关预测有待实验验证。
AI 中文摘要
塔耳博特效应（Talbot effect）的可观测量是光栅后方的强度，即演化场的密度，而非场本身。其分形性此前仅在带有有理跳跃的阶跃数据中被证实，本文证明了一般情况：对于任意有界变差且至少存在一个跳跃的实值数据，环面上自由薛定谔演化的密度几乎在所有时刻均为分形，其上盒维数恰好为3/2。其临界索伯列夫质量对数级发散，发散速率由数据的维纳跳跃统计量以闭式形式给出，与跳跃的位置和相位无关。该速率沿时间迹线减半，沿艾里流（Airy flow）加倍，因此临界质量可探测色散关系。该理论由无序熵算子（derangetropy operators）的秩演算推导而来，为每个概率分布赋予量子地毯：有理时刻分位数单元上的高斯和复兴，以及几乎所有其他时刻的普适分形。单个测量强度轮廓的谱增长率可返回光栅的跳跃内容，该预测有待在光学和物质波干涉测量中验证。
英文摘要
The observable of the Talbot effect is the intensity behind the grating: the density of the evolving field, not the field itself. Its fractality was previously known only for step data with rational jumps. We prove the general case. For arbitrary real data of bounded variation with at least one jump, the density of the free Schrödinger evolution on the torus is fractal at almost every time, with graph of upper box dimension exactly three halves. Its critical Sobolev mass diverges logarithmically, at a rate given in closed form by Wiener's jump statistic of the datum and independent of the positions and phases of the jumps. The same rate is halved along time traces and doubled for the Airy flow, so the critical mass probes the dispersion relation. Transported by the rank calculus of derangetropy operators, the theory equips every probability law with a quantum carpet: Gauss-sum revivals on quantile cells at rational times, a universal fractal at almost every other. The spectral growth rate of a single measured intensity profile returns the jump content of the grating, a prediction open to test in optical and matter-wave interferometry.
