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On a lattice point problem arising in the spectral analysis of periodic operators

Part of: Geometry of numbers

Published online by Cambridge University Press:  26 February 2010

Sergei V. Konyagin ,
Maxim M. Skriganov  and
Alexander V. Sobolev
Sergei V. Konyagin
Affiliation:
Moscow Lomonosov State University, Moscow, 119899, Russia, E-mail: [email protected]
Maxim M. Skriganov
Affiliation:
Centre for Mathematical Analysis and Its Applications, University of Sussex, Falmer, Brighton, BN1 9QH
Alexander V. Sobolev
Affiliation:
Department of Mathematics, University of Sussex, Falmer, Brighton. BN1 9RF, E-mail: [email protected]
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Abstract

Let N(ρ; ω) be the number of points of a d-dimensional lattice Γ. where d≥2, inside a ball of radius ρ centred at the point ω. Denote by (ρ) the number N(ρ; ω) averaged over ω in the elementary cell Ω of the lattice Γ. The main result is the following lower bound for for dimensions d ≅ l(mod 4):

MSC classification

Secondary: 11H06: Lattices and convex bodies
Type
Research Article
Information
Mathematika , Volume 50 , Issue 1-2 , December 2003 , pp. 87 - 98
Copyright
Copyright © University College London 2003

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References

1.Belov, A. S. and Konyagin, S. V.. On an estimate for the free term of a nonnegathe trigonometric polynomial with integer coefficients. Izv. Math. 60 (1996) 1123–1182.CrossRefGoogle Scholar
2.Dahlberg, B. E. J. and Trubowitz, E.. A remark on two dimensional periodic potentials. Comment. Math. Helvetici 57 (1982) 130–134.CrossRefGoogle Scholar
3.Erdélyi, A.. Higher Transcendental Functions, V. II, (McGraw-Hill, 1953).Google Scholar
4.Helffer, B. and Mohamed, A.. Asymptotics of the density of states for the Schrödinger operator with periodic electric potential, Duke Math. J. 92 (1998) 1–60.CrossRefGoogle Scholar
5.Krätzel, E.. Lattice Points (Kluwer, 1988).Google Scholar
6.Odlyzko, A. M.. Minima of cosine sums and maxima of polynomials on the unit circle. J. Lond. Math. Soc. 26, (1982) 412–420.CrossRefGoogle Scholar
7.Parnovski, L. and Sobolev, A. V.. On the Bethe-Sommerfeld conjecture for the polyharmonic operator, Duke Math. J. 107 (2001) 209–238.CrossRefGoogle Scholar
8.Parnovski, L. and Sobolev, A. V.. Lattice points, perturbation theory and the periodic polyharmonic operator, Ann. Henri Poincaré 2 (2001) 573–581.CrossRefGoogle Scholar
9.Skriganov, M. M.. Geometrical and Arithmetical Methods in the Spectral Theory of the Multidimensional Periodic Operators, Proc. Steklov Math. Inst. Vol. 171 (1984).Google Scholar
10.Skriganov, M. M.. The spectrum band structure of the three-dimensional Schrödinger operator with periodic potentials, Invent. Math. 80 (1985) 107–121.CrossRefGoogle Scholar