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On Riemann “nondifferentiable” function and Schrödinger equation. (English. Russian original) Zbl 1207.26010
Proc. Steklov Inst. Math. 269, 186-196 (2010); translation from Trudy Mat. Inst. Steklova 269, 193-203 (2010).
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 Schrödinger equation. The authors study some of the partial derivatives of \(\psi(x,t)\), as far as they show that the local Lipschitz-Hölder smoothness exponent of it in the variable \(t\) equals \(3/4\) almost everywhere on \(\mathbb R^2\).

MSC:
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
35Q41 Time-dependent Schrödinger equations and Dirac equations
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References:
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