# Publications / Research

## 2 Hyperbolic dynamical systems, Classical chaos

### 2.1 Introduction

This short text presents the list of papers given below, following a thematic description. This series of papers concern the hyperbolic dynamical systems, i.e. “chaotic dynamics” with high sensitivity to initial conditions, also called more precisely “Anosov dynamics”. The point of view here (following David Ruelle, R. Bowen and others), is to consider not evolution of individual trajectories that look like random, but the evolution of a probability distribution that appears to be more predictible. The tools are spectral analysis of the linear pull back operator: the flow (or the map) acting on functions (or sections of bundles) composition. Since chaotics dynamics generates oscillations with high frequencies that get larger and larger with time, it appears necessary to use microlocal analysis and symplectic geometry on cotangent bundle to study these operators.
• Motivation: the motivation to study these topics came after the study of quantum chaos (see below) that consists of quantization of some “chaotic classical dynamics”. In quantum chaos, the quantization is not unique in general (except for some algebraic dynamics as the linear cat map or geodesic flow on constant negative surfaces) so the questions of quantum chaos may seem to be not well defined. By studying directly the pull back operator of dynamical systems, the surprise has been to observe a “natural quantization” (and so unique).
• “Ruelle discrete spectrum”: In the paper (2.) we simply consider expanding map on ${S}^{1}$ and hyperbolic map on ${T}^{2}$, introducing the technics of “escape function” to construct “anisotropic Sobolev spaces” and reveal the “Ruelle discrete spectrum”. An other very simple “matrix model” that explains this approach is given in the appendix of paper (14.). In (3.), we show how to use microlocal analysis to extend this approach to general Anosov diffeomorphisms and to general Anosov flows in (5.). In (14.) we present a similar approach using a sligthly different technics of microlocal analysis (wave-packets transform), that is more geometric and allows to improve the escape function, and finally improve some bounds on the resolvent operator.
• “Natural quantization”, “Emergence of quantum dynamics in contact Anosov dynamics”: In the paper (1.) we study the Ruelle spectrum of the contact extension of a linear symplectic Anosov map on ${T}^{2}$ (also called prequantum cat map). This is a partially hyperbolic dynamical systems. We show that the long time behavior is governed by an effective operator isomorphic to the “quantum cat map”, i.e. quantum dynamics appears dynamically without an “ad hoc” quantization procedure as usual. In spectral terms, the Ruelle spectrum contains the quantum spectrum as an external band (or annulus). This phenomena exists more generally for the contact extension of any symplectic Anosov diffeomorphism (6.). Under some assumptions, this extension exists and is almost unique so intrinsic. This can also be done for contact Anosov flows 16. , announced in (7.). In (9.) we study a contact Anosov flow acting on a special line bundle $det{\left({E}_{s}\right)}^{1/2}$called “semiclassical bundle”, and shows that the first band of eigenvalues concentrates on the imaginary axis and are the zeroes of “the semiclassical zeta function of Gutzwiller-Voros”. This “semiclassical” zeta function somehow generalizes the Selberg zeta function (that is for constant negative curvature) to variable negative curvature and more general contact Anosov flows. In (10.) we consider the special case of hyperbolic manifolds (these are dynamical systems constructed from Lie groups as the geodesic flow on surfaces of constant negative curvature but in higher dimensions) and it is observed that the Ruelle spectrum is on vertical lines, using group theory. In (13.) we consider 3 dimensional contact Anosov flow and a different argument to show the band spectrum.
• “Partially hyperbolic dynamics”. The Ruelle spectrum of a simple model that is partially hyperbolic is studied in 4., i.e. there is a neutral direction for the dynamics. We use microlocal analysis and shows the existence of a spectral gap (so exponential decay of correlations) under generic assumptions. A similar study for open partially hyperbolic dynamics is done in (8.) and (11.). It is observed that neutral directions in general implies the existence of larger and larger number of Ruelle eigenvalues in the limit where the frequency in the neutral direction is high (called semiclassical limit). In the paper 12. we study the asymptotic spectral gap and try to improve the estimates using considerations after the Ehrenfest time..
• “Pseudo-Anosov dynamics”. In (15.) we study the Ruelle spectrum of linear pseudo Anosov map, i.e. there are singularities on the surface that are branching points.

## 3 Quantum chaos

### 3.1 Introduction

This short text presents the list of papers given below, following a thematic description. This series of papers concern “quantum chaos”, this means the dynamics of a linear PDE for which, in the limit of small wave length $h\ll 1$, called semiclassical limit, the Hamiltonian dynamics that describes motion of wave packets or rays, is chaotic, e.g. hyperbolic. In quantum chaos, there is a special time called Ehrenfest time ${t}_{E}$ defined as the typical time when small wave length $h$ are amplified to macroscopic size $\sim 1$ under the chaotic dynamics with Lyapunov exponent $\lambda >0$: this gives ${e}^{\lambda {t}_{E}}h\sim 1↔{t}_{E}\sim \frac{1}{\lambda }ln\left(\frac{1}{h}\right)$. Notice that even if $h$ is very small, ${t}_{E}$ is not very large due to $ln\left(\frac{1}{h}\right)$.
• The paper 1. study waves in a wave guide in the semiclassical limit of small wave lenghts. The conductivity of the wave guide (i.e. number of propagating modes) is related to ergodic properties of the rays in the guide.
• In papers 2. and 3. we study the eigenfunctions of the “quantum cat map”, i.e. the quantization of a hyperbolic linear symplectic map on the torus ${T}^{2}$. The surprise is to observe that, under some specific conditions, some eigenfunctions do not equidistribute as we could expect (and as it has been conjectured before under the name Quantum Unique Ergodicity, QUE). This shows the possible existence of “scars” in some uniform hyperbolic system (the only known example up to now) and gives a counter example to the “quantum unique ergodicity conjecture, QUE” in quantum chaos. One interesting feature in this model is the periodic behavior of wave packets: they spend some time to spread along the unstable direction of the dynamics and after some while they surprisingly reconstruct along the stable direction. The period $T$ of this phenomena is always larger than twice the Ehrenfest time, $T\ge 2{t}_{E}$, i.e. the regime where still mysterious interference phenomena are present in quantum chaos. Here are some movies.
• In paper (4.) we try to study long time behavior of wave packets in quantum chaos, i.e. long after the Ehrenfest time. This study has been started with the the “trace formula”. After this paper we have been convinced to consider first the dynamics of pull back operators (Ruelle spectrum, previous section) before continuing quantum chaos.

### 3.3 Lecture notes

• 2012. "Ergodicity of quantum maps". Manuscript lectures notes given in Marburg, september 2012. pdf file.
• 2005. "Long time semiclassical evolution of wave packets in quantum chaos. Example of non quantum unique ergodicity with hyperbolic maps" Lecture notes for I.H.P. School, june 2005. format ps.gz ou pdf
• 2007. "Introduction au chaos quantique"(Cours à l'école d'été de Peyresq 2007).
• 2014. “Introduction au chaos classique et quantique”(Notes d'exposé pour les journées mathématiques (journées X-UPS) à l'intention des professeurs de math de classes préparatoires, lundi 28 et mardi 29 avril 2014.)

## 4 General Quantum physics

### 4.2 Lecture notes

• 1993. "Approche géométrique de la limite semi-classique par les états cohérents", Exposé à grenoble 1993.
• 1997. "La quantification du champ électromagnétique" Exposé le juin 1997 à l'institut Fourier. (fichier ps).
• 2000. "La quantification géométrique" Exposé le 27 avril 2000 à l'institut Fourier.
• 2003. "Introduction à la mécanique quantique et au paradoxe E.P.R" (exposé au magistère de mathématiques, UJF, janvier 2003).
• 2017. "Exposé sur le théorème adiabatique en mécanique quantique. Description par l'analyse semiclassique". Exposé le 5 mai 2017 à l'institut Fourier.

## 5 Topological phenomena for slow-fast dynamical systems and molecular physics

### 5.1 Introduction

This short text presents the list of papers given below, following a thematic description. Before explaining each paper that corresponds to simple and specific examples we describe the abstract general phenomena.
##### Description of the general phenomena:
This series of papers concern some “topological phenomena in quantum mechanics”, and more precisely the manifestation of the Atiyah-Singer index formula in quantum physics of slow-fast dynamical systems also called adiabatic systems when a quantum physical system consists of sub-systems that interact together but each with different time scale. Example: small molecules (with few atoms) are quantum objects but quite complicated because electrons are light so move fast, they interact together but also with the atoms that are heavier so slower. These atoms have vibration motions that are still faster then the whole rotationnal motion of the molecule that is still slower. Geometricaly we can describe the global dynamics with different time scale as a bundle $F$ where the fast dynamics takes place in the fibers whereas the slow dynamics takes place on the base.
Of course an isolated molecule should be considered within a quantum description. Experimentally, by spectroscopy and fluorescence, physicists observe the energy levels (i.e. eigenvalues of the Hamiltonian quantum operator). They observe that energy levels of the molecules are discrete but form some groups or clusters (called energy bands). The exact number of energy levels in a cluster is given by the index formula that expresses this number in terms of the topology of a specific vector bundle $F$ that describes the dynamics of the molecule.
We give now more precise informations. As we have said, this clustering structure is due to the coexistence of motions of the molecules with different time of scale: the rotation of the whole molecule is slower than the faster oscillations of the atoms. In heuristic terms (from the uncertainty principle in Fourier transform), we have time scales ${T}_{rotation.}\gg {T}_{vibration.}$ that corresponds to separation of energy levels $\Delta {E}_{rot.}=\frac{\hslash }{{T}_{rot}}\ll \Delta {E}_{vib.}=\frac{\hslash }{{T}_{vib}}$. Using microlocal analysis we can describe with good accuracy the slow rotationnal motion using a classical mechanics i.e. dynamics on a symplectic manifold ${M}_{rot}$. The fast motion is described in quantum mechanics by a finite rank operator ${H}_{vib}\left(x\right)$ that depends on the point $x\in {M}_{rot}$. This gives a bundle of operators over ${M}_{rot.}$. Each eigenspace defines a finite rank vector bundle $F$ over ${M}_{rot}$. Each eigenspace $F$ corresponds to an observed cluster in the spectrum. The Atiyah-Singer index formula gives that the exact number of eigenvalues $N$ in a cluster is$N={\int }_{{M}_{rot}}Todd\left(T{M}_{rot}\right)\wedge Ch\left(F\otimes L\right)$where $Todd\left(T{M}_{rot}\right)$ is the Todd class of the tangent bundle of ${M}_{rot}$, $L$ is the line bundle over ${M}_{rot}$ whose curvature is the symplectic form ($L$ is called the prequantum line bundle) and $Ch\left(.\right)$ is the Chern class.
• A simple model with a vector bundle over ${S}^{2}$: The first appearance of this phenomena explained in (1.,2.,3.) was in the quantum spectrum of small molecules as $C{F}_{4}$. This a simple model where the dynamics of the molecule is described by a bundle of $2×2$ matrices over the symplectic sphere ${S}^{2}$. The eigenspaces form a rank 1 vector bundle over ${S}^{2}$ with Chern index $±C\in Z$ and the above index formula predicts (in confirmation we experimental observations) that the number of eigenvalues in a cluster is $N=J+1±C$ where $J$ is the angular momentum of the molecule.
• A model with a rank 2 vector bundle over ${CP}^{2}$ and surprising topological phenomena: In paper 4. we describe a realistic and more rich situation where the dynamics of the molecule is described by a rank 2 vector bundle over $C{P}^{2}$. In that case the Chern class is polynomial with integer coefficients and algebraic topology gives more subtil rules that are discussed, as topologically coupled bands, when the polynomial can not be factorized in the ring of polynomials with integer coefficients.
• Pedagogical presentation and manifestation of the Euclidean index formula in geophysics and quantum physics: In paper 6. we describe a similar phenomena that has been discovered in a different field of physics, namely geophysical waves (atmospheric or ocean surface waves near the equator). We also give a more pedagogical presentation (in the Arxiv version), and treat a very simple model in any dimension related to the Euclidean index formula of Fedosov-Hörmander.

### 5.2 Publications

1. 2000. "Topological Chern indices in molecular spectra" F. Faure, B. Zhilinskii, Physical Review Letters , 85, 960-963 , (2000). Paper, or preprint.
2. 2001. "Topological properties of the Born-Oppenheimer approximation and implications for the exact spectrum" F. Faure, B. Zhilinskii, Letters in Math. Physics vol 55, 219-238, (2001). Paper or preprint.
3. 2002. "Qualitative features of intra-molecular dynamics. What can be learned from symmetry and topology" F. Faure and B. Zhilinskií, Acta Appl. Math. 70, 265-282 (2002). Paper in ps.gz .
4. 2002. "Topologically coupled energy bands in molecules" F. Faure and B. Zhilinskií, Phys. Letters A, 302, p.242-252, 2002. Paper, or preprint.
5. 2007. "Adiabatically coupled systems and fractional monodromy" M. Hansen, F. Faure, B. Zhilinskii, J. Phys. A: Math. Theor. 40 (2007) 13075-13089. Paper of preprint
6. 2019. "Manifestation of the topological index formula in quantum waves and geophysical waves" F. Faure, preprint, to appear in AHL

## 6 Semi-classical and topological analysis of the quantum Hall effect

### 6.1 Publications

1. 1993. "Mécanique quantique sur le tore et dégénérescences dans le spectre". F. Faure, Séminaires d'analyse spectrale de l'Institut Fourier, Vol n°11, 1993. Paper.
2. 1994. "Generic description of the degeneracies in Harper like models." F. Faure, Journal of Physics A: Math. and gen. 27, (1994), 7519-7532. Paper.
3. 2000. "Semi-classical Quantum Hall conductivity" F. Faure, B.Parisse, Journal of Mathematical physics, 41, 62-75, (2000). Paper or preprint
4. 2000. "Topological properties of quantum periodic Hamiltonians" F. Faure, Journal of physics A: math and general, 33 , 531-555 (2000). Paper or preprint.

### 6.2 Lecture notes

• 1992. "Structure of wave functions on the torus characterized by a topological Chern index" in Proceedings to workshop in Trieste 1992. Paper.
• 2000. "Rôle des indices topologiques de Chern en physique du solide et physique moléculaire. Calculs dans des modèles simples." Notes de cours à l'école d'été interdisciplinaire MÉTHODES TOPOLOGIQUES ET GÉOMÉTRIQUES: APPLICATION AUX SYSTÈMES DYNAMIQUES PHYSIQUE, CHIMIE, BIOLOGIE. Dijon, 26-30 juin 2000. (fichier ps).
• 2002. "Geometric and topological aspects of slow and fast coupled dynamical systems in quantum and classical dynamics". Lectures notes for lectures given in:Saclay. Spth., march-april 2002, and M.A.S.I.E. Spring School, Warwick, march 2002. Lecture notes: format ps.gz ou pdf (or 2 pages / page: format ps.gz ou pdf) or html.
• 2003. "Topological indices in molecular spectra". Exposé en avril 2003 au Mathematical Science Research Institute (MSRI) Berkeley Vidéo et PDF.