Advanced course in classical and quantum chaos

Frédéric Faure January 9, 2021

Abstract A Course for the 2021-2022 Master 2 degree in fundamental mathematics, Math- ematical Physics in University Grenoble Alpes.

1 Introduction

Chaotic classical dynamics is the property of some deterministic dynamical systems to posses sensitivity to initial conditions. An important example in geometry is the geodesic ﬂow on negatively curved Riemannian manifolds. This is the Hamiltonian ﬂow of a free particle on the manifold. The purpose of dynamical system theory is to predict long time behavior of particles. Chaos implies here that a typical trajectory seems unpredictable and behaves as random. It is then preferable to consider evolution of probability distri- butions under the ﬂow. Then it appears that statistical properties of a typical trajectory can be computed and the evolution of probability distributions is predictable: it converges towards a uniform measure called equilibrium (this is the mixing property) and the tran- sient evolution is governed by an eﬀective Schrödinger equation (namely a wave equation in “quantum chaos”), whose discrete spectrum is called Ruelle-Pollicott resonances. We will describe this recent functional approach in dynamical systems theory that has given many interesting results in this ﬁeld and is still very active. Quantum chaos is the manifestation of chaotic dynamics in quantum dynamics, i.e. evolution of functions (or waves) under the Schrödinger equation when the correspond- ing classical mechanics is chaotic. The corresponding important example is the wave equa- tion on negatively curved Riemannian manifolds. The purpose of quantum chaos theory is to predict long time behavior of waves, equivalently properties of stationary waves, i.e. eigenfunctions and eigenvalues of the Schrödinger operator. The diﬃculty is interference phenomena that are complicated due to chaos. Some known results is the Theorem of “quantum ergodicity” that shows the equidistribution of most of eigenfunctions. Some important conjectures is the random matrix conjecture and the unique quantum ergod- icity conjecture: they claim that (for a generic system) all eigenfunctions equidistribute and behaves at high frequency as eigenfunctions of a universal model of random matrices. Quantum chaos theory is important in physics of waves (acoustic, electromagnetism, quan- tum mechanics etc) but also in mathematics, in group theory, in number theory where a

1 very conjectural manifestation would be statistics of random matrices in the distribution of prime numbers and in the zeros of the Riemann zeta function.

2 Mathematical themes

We will see and use:

• Diﬀerential geometry: manifolds, ﬁelds of tensors and diﬀerential forms. [16][11][13, 14, 15]

• Riemann and symplectic geometry to describe the geodesic ﬂow.[9][10, 12]

• Dynamical systems theory: properties of ergodicity and mixing. Entropy, topo- logical pressure.[1][2][8].

• Operators in functional analysis [3], Semi-groups [4]. Micro-local analysis using the wave-packet transform (FBI transform, geometric quantization): this is a way to map isometrically functions or distribution on the manifold to functions on the cotangent bundle. Sobolev spaces.[17]

3 Contents

• Classical mechanics, deterministic chaos:

– Some example and properties of chaotic (i.e. hyperbolic, Anosov dynamics) dynamical systems: linear hyperbolic automorphism on the torus (Arnold’s cat map), dispersive billiards, geodesic ﬂow on negatively curved manifold, Lorenz ﬂow. – The classical evolution of probability distributions using micro-local analysis. Ruelle-Pollicott discrete spectrum that describes the long time evolution. The Atiyah-Bott trace formula. Selberg and Ruelle zeta functions that relates the Ruelle-Pollicott spectrum to the periodic orbits and give informations on the distribution of periodic orbits. Analogy with the distribution of primes. [7,6]

• Quantum mechanics, quantum chaos:

– within the framework of geometric quantization. A ﬁrst result is the theo- rem of propagation of singularities that shows that quantum wave packets evolves as particles for any time but in the limit of high frequencies, called the semi-classical limit. – Some properties of quantum chaos that can be deduced from short or ﬁnite time analysis versus frequency: the Weyl law, the theorem of quantum ergodicity. The trace formula of Duistermaat-Guillemin that relates the

2 spectrum and periodic orbits. Some conjectures related to long time analysis: random waves and random matrix conjectures.[5].

References

[1] V.I. Arnold. Les méthodes mathématiques de la mécanique classique. Ed. Mir. Moscou, 1976. [2] M. Brin and G. Stuck. Introduction to Dynamical Systems. Cambridge University Press, 2002. [3] E.B. Davies. Linear operators and their spectra. Cambridge University Press, 2007. [4] K.J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194. Springer, 1999. [5] F. Faure. Introduction au chaos quantique. In journées X-UPS, Éditions de l’école polytechnique link. 2014. [6] F. Faure. From classical chaos to quantum chaosspectrum. In link, Lectures notes for the school 22-26 April 2019 at CIRM. 2018. [7] F. Faure. Spectrum, traces and zeta functions in hyperbolic dynamics. In link, school 23-27 April 2018 at the University Cheikh Anta Diop in Dakar, Sénégal. 2018. [8] A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Sys- tems. Cambridge University Press, 1995. [9] Lee. Riemannian Manifolds:An Introduction to Curvature. Springer, 1997. [10] D McDuﬀ and D Salamon. Introduction to symplectic topology, 2nd edition. clarendon press, Oxford, 1998. [11] M. Nakahara. Geometry, topology and physics. Institute of Physics Publishing, 2003. [12] A. Cannas Da Salva. Lectures on Symplectic Geometry. Springer, 2001. [13] M. Taylor. Partial diﬀerential equations, Vol I. Springer, 1996. [14] M. Taylor. Partial diﬀerential equations, Vol II. Springer, 1996. [15] M. Taylor. Partial diﬀerential equations, Vol III. Springer, 1996. [16] M. Dillard-Bleick Y. Choquet-Bruhat, C. Dewitt-Morette. Analysis, manifolds and physics. North-Holland, 1982. [17] M. Zworski. Semiclassical Analysis. Graduate Studies in Mathematics Series. Amer Mathematical Society, 2012.

3