an:05816205
Zbl 1207.26010
Oskolkov, K. I.; Chakhkiev, M. A.
On Riemann ``nondifferentiable'' function and Schr??dinger equation
EN
Proc. Steklov Inst. Math. 269, 186-196 (2010); translation from Trudy Mat. Inst. Steklova 269, 193-203 (2010).
00267924
2010
j
26A27 35Q41
Riemann's nowhere-differentiable function; Schr??dinger equation; local Lipschitz-H??lder smoothness exponent
For \(\{x,t\}\in\mathbb R^2\), the authors consider the function
\[
\psi(x,t)=\sum_{n\in\mathbb Z\setminus\{0\}} \frac{e^{\pi i(tn^2+2xn)}}{\pi in^2},
\]
which is a generalization of Riemann's nowhere-differentiable function, and also it is a generalized solution of the Cauchy initial value problem for the Schr??dinger equation. The authors study some of the partial derivatives of \(\psi(x,t)\), as far as they show that the local Lipschitz-H??lder smoothness exponent of it in the variable \(t\) equals \(3/4\) almost everywhere on \(\mathbb R^2\).
Mehdi Hassani (Zanjan)