General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Trudy MIAN:

Personal entry:
Save password
Forgotten password?

Tr. Mat. Inst. Steklova, 2010, Volume 269, Pages 193–203 (Mi tm2905)  

This article is cited in 2 scientific papers (total in 2 papers)

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 $\mathbb R^2$.

Full text: PDF file (239 kB)
References: PDF file   HTML file

English version:
Proceedings of the Steklov Institute of Mathematics, 2010, 269, 186–196

Bibliographic databases:

UDC: 517.51+511.3
Received in February 2010

Citation: K. I. Oskolkov, M. A. Chakhkiev, “On Riemann “nondifferentiable” function and Schrödinger equation”, Function theory and differential equations, Collected papers. Dedicated to Academician Sergei Mikhailovich Nikol'skii on the occasion of his 105th birthday, Tr. Mat. Inst. Steklova, 269, MAIK Nauka/Interperiodica, Moscow, 2010, 193–203; Proc. Steklov Inst. Math., 269 (2010), 186–196

Citation in format AMSBIB
\by K.~I.~Oskolkov, M.~A.~Chakhkiev
\paper On Riemann ``nondifferentiable'' function and Schr\"odinger equation
\inbook Function theory and differential equations
\bookinfo Collected papers. Dedicated to Academician Sergei Mikhailovich Nikol'skii on the occasion of his 105th birthday
\serial Tr. Mat. Inst. Steklova
\yr 2010
\vol 269
\pages 193--203
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2010
\vol 269
\pages 186--196

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. K. I. Oskolkov, M. A. Chahkiev, “Traces of the discrete Hilbert transform with quadratic phase”, Proc. Steklov Inst. Math., 280 (2013), 248–262  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. Erdogan M.B. Shakan G., “Fractal Solutions of Dispersive Partial Differential Equations on the Torus”, Sel. Math.-New Ser., 25:1 (2019), UNSP 11  crossref  mathscinet  isi  scopus
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
    Number of views:
    This page:605
    Full text:48

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020