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On Riemann “nondifferentiable” function and Schrödinger equation
K. I. Oskolkova,
M. A. Chakhkievb a Department of Mathematics, University of South Carolina, Columbia, USA
b Russian State Social University, Moscow, Russia
Abstract:
The function $\psi:=\sum_{n\in\mathbb Z\setminus\{0\}}e^{\pi i(tn^2+2xn)}/(\pi in^2)$,
$\{t,x\}\in\mathbb R^2$, is studied as a (generalized) solution of the Cauchy initial value problem for the Schrödinger equation. The real part of the restriction of
$\psi$ on the line
$x=0$, that is, the function $R:=\operatorname{Re}\psi|_{x=0}=\frac2\pi\sum_{n\in\mathbb N}\frac{\sin\pi n^2t}{n^2}$,
$t\in\mathbb R$, was suggested by B. Riemann as a plausible example of a continuous but nowhere differentiable function. The points are established on
$\mathbb R^2$ where the partial derivative
$\frac{\partial\psi}{\partial t}$ exists and equals
$-1$. These points constitute a countable set of open intervals parallel to the
$x$-axis, with rational values of
$t$. Thereby a natural extension of the well-known results of G. H. Hardy and J. Gerver is obtained (Gerver established that the derivative of the function
$R$ still does exist and equals
$-1$ at each rational point of the type
$t=\frac aq$ where both numbers
$a$ and
$q$ are odd). A basic role is played by a representation of the differences of the function
$\psi$ via Poisson's summation formula and the oscillatory Fresnel integral. It is also proved that the number
$\frac34$ is the sharp value of the Lipschitz–Hölder exponent of the function
$\psi$ in the variable
$t$ almost everywhere on
$\mathbb R^2$.
UDC:
517.51+511.3
Received in February 2010
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