Fractality of the Schrödinger Density for Rough Data

Abstract

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.