On Riemann ``nondifferentiable'' function and Schr\"odinger equation

K. I. Oskolkov ; M. A. Chakhkiev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 193-203

Résumé

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 $\mathbb R^2$.