Abstract
In these lectures I describe in detail the quantization of linear symplectic maps of the torus, as a continuation of De Bievre’s lectures (De Bi`evre, 2006). I will then survey the problem of quantum equidistribution for this model. This model was introduced by Hannay and Berry (Hannay and Berry, 1980). It turns out that it has a rich arithmetic structure, and its study uses several ingredients in modern number theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bourgain, J. (2006) A remark on quantum ergodicity for cat maps, preprint.
Bouzouina, A. and De Bièvre, S. (1996) Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Comm. Math. Phys. 178, 83–105.
Colin de Verdière, Y. (1985) Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102, 497–502.
De Bièvre, S. (2006) An introduction to quantum equidistribution, in this book.
Degli Esposti, M., Graff, S., and Isola, S. (1995) Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math Phys. 167, 471–507.
Faure, F., Nonnenmacher, S., and De Bièvre, S. (2003) Scarred eigenstates for quantum cat maps of minimal periods, Comm. Math. Phys. 239, 449–492.
Friedlander, J. (2006) Uniform distribution, exponential sums and crypography, in this book.
Gurevich, S. and Hadani, R. (2003) The two dimensional Hannay—Berry model, arXiv:mathph/0312039.
Gurevich, S. and Hadani, R. (2006) Proof of the Kurlberg—Rudnick Rate Conjecture, C. R. Math. Acad. Sci. Paris 342, 69–72.
Hannay, J. H. and Berry, M. V. (1980) Quantization of linear maps on a torus - Fresnel diffraction by a periodic grating, Physica D 1, 267–291.
Kelmer, D. (2005) Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus, arXiv:math-ph/0510079.
Kurlberg, P. (2003) On the order of unimodular matrices modulo integers, Acta Arith. 110, 141–151.
Kurlberg, P. and Rudnick, Z. (2000) Hecke theory and equidistribution for the quantization of linear maps of the torus, Duke Math. J. 103, 47–78.
Kurlberg, P. and Rudnick, Z. (2001) On quantum ergodicity for linear maps of the torus, Commun. Math. Phys. 222, 201–227.
Lindenstrauss, E. (2006) Three examples how to use measure classification in number theory, in this book.
Mezzadri, F. (2002) On the multiplicativity of quantum cat maps, Nonlinearity 15, 905–922.
Rudnick, Z. and Sarnak, P. (1994) The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math Phys. 161, 195–213.
Schnirelman, A. (1974) Ergodic properties of eigenfunctions, Usp. Math. Nauk 29, 181–182.
Zelditch, S. (1987) Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55, 919–941.
Zelditch, S. (1996) Quantum ergodicity of C*-dynamical systems, Comm. Math.Phys. 177, 507–528.
Zelditch, S. (1997) Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47, 305–363.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this paper
Cite this paper
Rudnick, Z. (2007). THE ARITHMETIC THEORY OF QUANTUM MAPS. In: Granville, A., Rudnick, Z. (eds) Equidistribution in Number Theory, An Introduction. NATO Science Series, vol 237. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5404-4_15
Download citation
DOI: https://doi.org/10.1007/978-1-4020-5404-4_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-5402-0
Online ISBN: 978-1-4020-5404-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)