Konstantin Oskolkov

Abstract:   In the talk, self-similarity properties of the solutions of Schroedinger type equations with the periodic initial data functions will be discussed. The focus will be on the density generated by the solution of the free particle equation, with the Jacobi's theta function as the initial data. The 3d graph is a rather rugged landscape, which is criss-crossed by a set of deep rectilinear canyons, fairly well-organized (the valleys of shadow). Theorems explaining this peculiarity will be presented.
Key-words applied by physicists: Talbot's effect, quantum carpets, fractional and fractal revivals (Berry, Schleich, and many others).
      Self-similarity effects have been well known to mathematicians, and multiply utilized, since the creation of the circle method (Hardy, Littlewood, Ramanujan, Vinogradov, Weyl). Gauss' sums, or complete rational exponential sums of higher order, are scaling factors, while the oscillatory integrals with the polynomial phase play the role of the patterns of the arising ``arithmetical carpets."
        In optics, the multiscaled reproduction of the original periodic image (planar grating) has been experimentally discovered in 1836 by W.H.F. Talbot, the British inventor of photography.