Renormalization of exponential sums and matrix cocycles
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2004-2005), Exposé no. 16, 10 p.

Dans cet article, nous présentons un nouveau point de vue sur la renormalisation de certaines sommes exponentielles issues de la théorie des nombres. Nous généralisons cette procédure pour étudier certains cocycles matriciels liés à des problèmes spectraux de la mécanique quantique.

In this paper, we present a new point of view on the renormalization of some exponential sums stemming from number theory. We generalize this renormalization procedure to study some matrix cocycles arising in spectral problems of quantum mechanics

Classification : 34E05, 34E20, 34L05, 34L40
Mots clés : exponential sums, matrix cocycles, monodromy matrix
Fedotov, Alexander 1 ; Klopp, Frédéric 2

1 Departement of Mathematical Physics, St Petersburg State University, 1, Ulianovskaja, 198904 St Petersburg-Petrodvorets, Russia
2 LAGA, Institut Galilée, U.R.A 7539 C.N.R.S, Université de Paris-Nord, Avenue J.-B. Clément, F-93430 Villetaneuse, France
@article{SEDP_2004-2005____A16_0,
     author = {Fedotov, Alexander and Klopp, Fr\'ed\'eric},
     title = {Renormalization of exponential sums and matrix cocycles},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:16},
     pages = {1--10},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2004-2005},
     mrnumber = {2182060},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_2004-2005____A16_0/}
}
TY  - JOUR
AU  - Fedotov, Alexander
AU  - Klopp, Frédéric
TI  - Renormalization of exponential sums and matrix cocycles
JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:16
PY  - 2004-2005
SP  - 1
EP  - 10
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/item/SEDP_2004-2005____A16_0/
LA  - en
ID  - SEDP_2004-2005____A16_0
ER  - 
%0 Journal Article
%A Fedotov, Alexander
%A Klopp, Frédéric
%T Renormalization of exponential sums and matrix cocycles
%J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
%Z talk:16
%D 2004-2005
%P 1-10
%I Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://www.numdam.org/item/SEDP_2004-2005____A16_0/
%G en
%F SEDP_2004-2005____A16_0
Fedotov, Alexander; Klopp, Frédéric. Renormalization of exponential sums and matrix cocycles. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2004-2005), Exposé no. 16, 10 p. http://www.numdam.org/item/SEDP_2004-2005____A16_0/

[1] M. V. Berry and J. Goldberg. Renormalisation of curlicues. Nonlinearity, 1(1):1–26, 1988. | MR | Zbl

[2] V. Buslaev and A. Fedotov. On the difference equations with periodic coefficients. Adv. Theor. Math. Phys., 5(6):1105–1168, 2001. | MR | Zbl

[3] V. S. Buslaev. Kronig-Penney electron in a homogeneous electric field. In Differential operators and spectral theory, volume 189 of Amer. Math. Soc. Transl. Ser. 2, pages 45–57. Amer. Math. Soc., Providence, RI, 1999. | MR | Zbl

[4] V. S. Buslaev and A. A. Fedotov. Bloch solutions for difference equations. Algebra i Analiz, 7(4):74–122, 1995. | MR | Zbl

[5] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ. Ergodic theory, volume 245 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. | MR | Zbl

[6] E. A. Coutsias and N. D. Kazarinoff. The approximate functional formula for the theta function and Diophantine Gauss sums. Trans. Amer. Math. Soc., 350(2):615–641, 1998. | MR | Zbl

[7] J.-M. Deshouillers. Geometric aspect of Weyl sums. In Elementary and analytic theory of numbers (Warsaw, 1982), volume 17 of Banach Center Publ., pages 75–82. PWN, Warsaw, 1985. | MR | Zbl

[8] R. Evans, M. Minei, and B. Yee. Incomplete higher-order Gauss sums. J. Math. Anal. Appl., 281(2):454–476, 2003. | MR | Zbl

[9] M. V. Fedoryuk. Asymptotic analysis. Springer-Verlag, Berlin, 1993. Linear ordinary differential equations, Translated from the Russian by Andrew Rodick. | MR | Zbl

[10] A. Fedotov and F. Klopp. The multifractal structure of the generalized eigenfunctions and the spectral measure for an ergodic family of Schrödinger operators. In preparation.

[11] A. Fedotov and F. Klopp. Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case. Comm. Math. Phys., 227(1):1–92, 2002. | MR | Zbl

[12] G. H. Hardy and J. E. Littlewood. Some problems of Diophantine approximation. Acta Math., 37:193–238, 1914. | MR

[13] M. Mendès France. The Planck constant of a curve. In Fractal geometry and analysis (Montreal, PQ, 1989), volume 346 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 325–366. Kluwer Acad. Publ., Dordrecht, 1991. | MR | Zbl

[14] L.J. Mordell. The approximate functional formula for the theta function. J. London Mat. Soc., 1:68–72, 1926.

[15] J.G. van der Corput. Über Summen die mit den Elliptischen θ-Funktionen zusammenhägen I. Math. Ann., 87:66–77, 1922. | MR

[16] J.G. van der Corput. Über Summen die mit den Elliptischen θ-Funktionen zusammenhägen II. Math. Ann., 90:1–18, 1923.