## 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
Full Text:

### References:

 [1] K. Weierstrass, ”Über continuirliche Funktionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen,” in Mathematische Werke (Mayer & Müller, Berlin, 1895), Vol. 2, pp. 71–74. [2] G. H. Hardy, ”Weierstrass’s Non-differentiable Function,” Trans. Am. Math. Soc. 17, 301–325 (1916). · JFM 46.0401.03 [3] J. Gerver, ”The Differentiability of the Riemann Function at Certain Rational Multiples of {$$\pi$$},” Am. J. Math. 92, 33–55 (1970). · Zbl 0203.05904 [4] J. Gerver, ”More on the Differentiability of the Riemann Function,” Am. J. Math. 93, 33–41 (1971). · Zbl 0228.26008 [5] J. L. Gerver, ”On Cubic Lacunary Fourier Series,” Trans. Am. Math. Soc. 355, 4297–4347 (2003). · Zbl 1031.42003 [6] P. L. Butzer and E. L. Stark, ”’Riemann’s Example’ of a Continuous Nondifferentiable Function in the Light of Two Letters (1865) of Cristoffel to Prym,” Bull. Soc. Math. Belg., Sér A 38, 45–73 (1986). · Zbl 0629.01013 [7] J. J. Duistermaat, ”Self-similarity of ’Riemann’s Nondifferentiable Function’,” Nieuw Arch. Wisk., Ser. 4, 9, 303–337 (1991). · Zbl 0760.26009 [8] M. Holschneider and P. Tchamitchian, ”Pointwise Analysis of Riemann’s ’Nondifferentiable’ Function,” Invent. Math. 105, 157–175 (1991). · Zbl 0741.26004 [9] S. Itatsu, ”Differentiability of Riemann’s Function,” Proc. Jpn. Acad., Ser. A 57, 492–495 (1981). · Zbl 0501.26004 [10] S. Jaffard, ”The Spectrum of Singularities of Riemann’s Function,” Rev. Mat. Iberoam. 12, 441–460 (1996). · Zbl 0889.26005 [11] S. Jaffard and Y. Meyer, Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions (Am. Math. Soc., Providence, RI, 1996), Mem. AMS 123, No. 587. · Zbl 0873.42019 [12] W. Luther, ”The Differentiability of Fourier Gap Series and ’Riemann’s Example’ of a Continuous, Nondifferentiable Function,” J. Approx. Theory 48, 303–321 (1986). · Zbl 0626.42008 [13] E. Mohr, ”Wo ist die Riemannsche Funktion · Zbl 0456.26003 [14] E. Neuenschwander, ”Riemann’s Example of a Continuous, ’Nondifferentiable’ Function,” Math. Intell. 1, 40–44 (1978). · Zbl 0374.26002 [15] H. Queffelec, ”Dérivabilité de certaines sommes de séries de Fourier lacunaires,” C. R. Acad. Sci. Paris, Sér. A 273, 291–293 (1971). · Zbl 0221.26008 [16] K. I. Oskolkov, ”A Class of I.M. Vinogradov’s Series and Its Applications in Harmonic Analysis,” in Progress in Approximation Theory: An International Perspective (Springer, New York, 1992), pp. 353–402. · Zbl 0815.42003 [17] K. I. Oskolkov, ”Vinogradov Series in the Cauchy Problem for Equations of Schrödinger Type,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 200, 265–288 (1991) [Proc. Steklov Inst. Math. 200, 291–315 (1993)]. · Zbl 0831.35038 [18] K. I. Oskolkov, ”Series and Integrals of I.M. Vinogradov and Their Applications,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 190, 186–221 (1989) [Proc. Steklov Inst. Math. 190, 193–229 (1992)]. · Zbl 0707.11059 [19] K. I. Oskolkov, ”The Schrödinger Density and the Talbot Effect,” in Approximation and Probability (Inst. Math., Pol. Acad. Sci., Warsaw, 2006), Banach Center Publ. 72, pp. 189–219. · Zbl 1140.42300 [20] G. I. Arkhipov and K. I. Oskolkov, ”On a Special Trigonometric Series and Its Applications,” Mat. Sb. 134(2), 147–157 (1987) [Math. USSR, Sb. 62 (1), 145–155 (1989)]. · Zbl 0665.42003 [21] K. I. Oskolkov, ”On Functional Properties of Incomplete Gaussian Sums,” Can. J. Math. 43, 182–212 (1991). · Zbl 0728.11039 [22] A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959). · Zbl 0085.05601 [23] A. Ya. Khinchin, Continued Fractions (Fizmatgiz, Moscow, 1961; Univ. Chicago Press, Chicago, 1964). [24] K. J. Falconer, The Geometry of Fractal Sets (Cambridge Univ. Press, Cambridge, 1985). · Zbl 0587.28004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.