We study the local properties of the time-dependent probability density function for the free quantum particle in a box, i.e. the squared magnitude of the solution of the Cauchy initial value problem for the Schrödinger equation with zero potential, and the periodic initial data. √δ-families of initial functions are considered whose squared magnitudes approximate the periodic Dirac δ-function. The focus is on the set of rectilinear domains where the density has a special character, in particular, remains bounded, or even has low average values ("the valleys of shadows").
An essential part of the paper is dedicated to a review of some earlier results concerning the fractal properties of Vinogradov's extensions, which incorporate the solutions of a wide class of Schrödinger type equations. Relations are discussed with the optical diffraction phenomena discovered in 1836 by W. H. F. Talbot, the British inventor of photography. In the modern Physics literature, self-similarity in the wave diffracted by periodic gratings, is known as fractional and fractal revivals, and quantum carpets (M. Berry, W. Schleich, and many others). Self-similarity has been well-known, and extensively utilized in Analytic Number Theory, since the creation of the circle method of Hardy-Littlewood-Ramanujan, and Vinogradov's method of estimation and asymptotic formulas for H. Weyl's exponential sums. According to these methods, on the major arcs, the complete rational exponential sums are the scaling factors, while the appropriate oscillatory integrals constitute the pattern of the arising arithmetical carpets.