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The valleys of shadow in Schrödinger landscape


A 2003 Preprint by K. Oskolkov

  • 2003:11
  • The probability density function is studied for the one-dimensional quantum particle whose motion is defined by the Schrodinger equation

    $$\frac{\delta\psi}{\delta{t}}=\frac{1}{2\pi{i}}\frac{\delta^2\psi}{\delta{x^2}},\,\,\,\,\psi(f;t,x)\Big | _ {t=0}=f(x),$$

    with the periodic initial data $f,\ f(x+1)\equiv{f(x)}$. For f of the type$f _ \varepsilon(x):=c(\varepsilon)e^{-\frac{\langle{x}\rangle^2}{\varepsilon}},$ $\varepsilon-$ a small positive parameter, $\langle{x}\rangle-$ the distance from x to the nearest integer, Daniel Dix conducted a numerical experiment of 3d-graphing the density $|\psi(f _ \varepsilon;t,x)|^2$. Visually, the graph resembled a mountain landscape scarred by a peculiar discrete collection of deep rectilinear canyons, or "the valleys of shadow". We prove that this phenomenon is common for a wide set of families of the initial data $\{f _ \varepsilon\}$ such that the initial densities $\{|f _ \varepsilon|^2\}$ approximate, as $\varepsilon\to0,$ the periodic Dirack's delta-function: the Radon transformations of $|\psi(f _ \varepsilon)|^2$ are indeed small on a definite collection of lines on the plane $(t,x)$. A complete description of such collections is established, and applications to Hemholtz equation are discussed.

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