Workshop at CIRM Luminy, France

January 19-23, 2009

"Resonances in physics and mathematics"

This conference is part of the program "Resonances and decoherence in quantum chaos" financed by the French research agency ANR. The conference receives additional support from CIRM (the venue) and participation of C.M.L.S. (Centre de Mathématiques Laurent Schwartz).

ORGANIZING COMMITTEE: Nalini Anantharaman, Frédéric Faure et Stéphane Nonnenmacher.

SCIENTIFIC COMMITTEE: Nalini Anantharaman (Ecole Polytechnique), Viviane Baladi (Paris), Yves Colin de Verdière (Grenoble),Frédéric Faure (Grenoble), Colin Guillarmou (Nice), Frédéric Naud (Avignon), Stéphane Nonnenmacher (CEA Saclay), Dominique Spehner (Grenoble)

Presentation

Schedule, Program

Presentation

The main subject of this workshop is resonances as they appear in many domains of mathematics and theoritical physics. From a general point of view, "resonances" comes from a spectral description applied to dynamical problems. The notion of "resonances" has been developped over the last few years and appears under various aspects with different interpretations, depending on the domains of mathematics or physics. We would like that these different communities meet together, confront and share there approachs, knowledges and technics:

Resonances in experimental physics.
Quantum resonances.

Resonances in open quantum systems. Decoherence.
Resonances in semi-classical systems. Quantum chaos.
Resonances in hyperbolic geometry.
Ruelle Pollicott resonances of hyperbolic dynamical systems or partially hyperbolic.

Présentation détaillée (in French).

Invited Speakers

Experimentalists

Ulrich Kuhl (Marburg)

"Studies of resonances in open microwave cavities by the method of harmonic inversion"
From the measurement of a reflection spectrum of an open microwave cavity the poles of the scattering matrix in the complex plane have been determined [1]. The resonances have been extracted by means of the harmonic inversion method [2]. By this it became possible to resolve the resonances in a regime where the line widths exceed the mean level spacing up to a factor of 10, a value inaccessible in experiments up to now. For example the distributions of line widths were studied and found to be in good agreement with predictions from random matrix theory [3].
[1] UK, R. Höhmann, J. Main, and H.-J. Stöckmann, Phys. Rev. Lett. 100, 254101 (2008)
[2] J. Main, Phys. Rep. 316, 233 (1999).
[3] H.~J. Sommers, Y.~V. Fyodorov, and M. Titov, J. Phys. A 32, L77 (1999).
Fabrice Mortessagne (Nice)

" Wave Chaos in a chaotic optical fiber: from ergodic modes to gain-controlled scar modes"
For now a few years, we experimentally proved that a passive optical fiber with a chaotic-billiard-shape transverse section constitutes a powerful system for studying (and imaging) wave chaos [1], and particularly when scars are involved [2]. Recently, we propose an original mechanism to select scar modes through coherent gain amplification in a multimode D-shaped fiber. More precisely, we demonstrate the selective amplification of scar modes by positioning a gain region in the vicinity of the self-focal point of the shortest periodic orbit in the transverse motion.
[1] V. Doya, O. Legrand and F. Mortessagne, Phys. Rev. E 65, 056223 (2002).
[2] V. Doya, O. Legrand and F. Mortessagne, Phys. Rev. Lett. 88, 014102 (2002).
[3] C. Michel, V. Doya, O. Legrand and F. Mortessagne, Phys. Rev. Lett. 99, 224101 (2007).
Mélanie Lebental (Paris-Nord)

"Organic micro-lasers as test-beds for wave chaos in open systems" Slides
There is currently a great deal of interest in flat dielectric micro-cavities due to their numerous practical applications and their use to test fundamental physics such as quantum chaos. Actually, due to a formal analogy, the electromagnetic field plays the role of a quantum particule. The classical limit then corresponds to the geometrical optics limit. The original aspect of our study is based on the use of organic materials to make flat micro-lasers with various shapes (stadium, disk, polygons, etc...). In fact their low refractive indexes facilitate the output coupling of the light confined inside the resonator. The generic features of these open billiards are very different from what is expected for equivalent closed systems. In particular the trace formula must be modified to take into account the losses. It can be derived analytically in two cases and we proposed a general expression which can be tested experimentally. We demonstrated that lasing can be used to monitor the weight coefficients of the periodic orbits and get an insight into the imaginary part of the passive resonator spectrum.
Achim Richter (Darmstadt)

"Chaotic Scattering in Microwave Billiards: Isolated and Overlapping Resonances*" Slides
Measurements of transmission and reflection spectra of two-dimensional microwave resonators coupled to two antennas are discussed. Below a certain cut-off frequency the resonators simulate a chaotic quantum system whose eigenvalues manifest themselves as resonances with average spacing D and average width Γ. Experimental observables are scattering amplitudes, i.e. S-matrix elements, and their phases determined in the region of isolated and overlapping resonances for time-reversal invariant (GOE) and non-invariant (GUE) systems, respectively, as a function of frequency. Particular emphasis is given to the region of overlapping resonances, i.e. Γ ≥ D, where two S-matrix elements are correlated over a certain range of frequency. If, however, the S-matrix elements are Fourier transformed from the frequency into the time domain a sample of statistically independent Fourier coefficients is obtained which can then be compared to predictions from the theory of chaotic scattering developed in the framework of nuclear reaction theory [1,2,3]. One of the results discussed is: although a single resonance decays exponentially in time the decay of a system of resonances with random parameters is in general non-exponential.
* Supported by the DFG within the SFB634
[1] H. Alt, H.-D. Gräf, H.L. Harney, R. Hofferbert, H. Lengeler, A. Richter, P. Schardt, H.A. Weidenmüller, Phys. Rev. Lett. 74,62 (1995).
[2] B.Dietz, T. Friedrich, H.L. Harney, M. Miski-Oglu, A. Richter, F. Schäfer, H.A. Weidenmüller, Phys. Rev. E, in press.
[3] B.Dietz, T. Friedrich, H.L. Harney, M. Miski-Oglu, A. Richter, F. Schäfer, J.J.M. Verbaarshoot, H.A. Weidenmüller, to be published.

Theoretical physicists

Dmitry Savin (Brunel, West London)

"Statistics of quantum resonances and fluctuations in chaotic scattering" Slides
Resonances in (partly) open systems are quasi-bound states embedded in continuum. A natural way to address them is via the S-matrix whose poles turn out to be complex eigenvalues of an effective non-Hermitian Hamiltonian. We consider and overview their universal spectral properties in the chaotic regime as given by an appropriate non-Hermitian RMT. The most salient manifestations of the width fluctuations in scattering are also discussed and illustrated by non-exponential decay law and complexness (bi-orthogonality) of resonance states.
Henning Schomerus (Lancaster)

"Lifetime distribution in open systems: beyond ballistic chaotic decay" Slides
In order to discuss resonances in very open systems (such as dielectric microresonators) it is desirable to adopt concepts which have been successfully applied to other classes of open systems, such as Weyl laws and random-matrix theory (RMT). This talk focuses on a class of dynamical models [1,2,3] which allows to address the distribution of resonances in systems which are very open [3], where the decay is not ballistic [3,4,5], or where the internal dynamics is not fully chaotic [4]. Results from individual systems (quantum maps [1-4] and the dielectric stadium resonator [5]) are compared to the predictions of a renormalized RMT, where the renormalization in some cases corresponds to a fractal Weyl law.
[1] H. Schomerus and J. Tworzydlo, Phys. Rev. Lett. 93, 154102 (2004)
[2] H. Schomerus and P. Jacquod, J. Phys. A: Math. Gen. 38, 10663-10682 (2005)
[3] J. P. Keating, M. Novaes, and H. Schomerus, Phys. Rev. A 77, 013834 (2008)
[4] M. Kopp and H. Schomerus, in preparation
[5] H. Schomerus, J. Wiersig, and J. Main, in preparation
Niels Sondergaard (Lund)

"Scattering combined with quantum chaos"
We introduce our methodology quantum chaos by discussing experiments on elastic resonators. This leads to spectral densities, Weyl counting functions and periodic orbits. When turning to the exterior problem, semiclassical expansions of scattering determinants produce quantum zeta functions. One feature shared with their cousins in number theory is the Euler product over prime elements, here prime orbits trapped in the scattering geometry. Depending on the underlying linear wave equation, these orbits can take several forms. In the elastodynamic setting, this is indeed the case due to polarization and surface orbits. Using scattering, we will also remark on various trace formulae for the interior problem.
Jan Wiersig (Magdeburg, Germany)

"Quantum chaos in optical microcavities" Slides
Dielectric optical microcavities are important for a wide range of research areas and applications, such as ultra-low threshold lasers and single-photon sources. In the nonlinear dynamics and quantum chaos community microdisk cavities with deformed cross-sectional shape attracted considerable attention since they can be used to study the ray-wave correspondence in open systems in direct comparison to experiments. In this talk we discuss the following aspects: (i) fractal Weyl law, (ii) avoided resonance crossings, and (iii) the influence of the chaotic repeller on the far-field emission pattern of microcavities.

Quantum systems coupled to a bath

Laurent Bruneau (Cergy)

"Random repeated interaction quantum systems" Slides
We consider a small quantum system S interacting in a successive way with a sequence E_1,E_2,... of independent quantum subsystems, so-called "repeated interaction systems". In an ideal situation all the interactions are identical: all the subsystems E_m are the same, as well as the duration of each interaction. In this talk, we consider the situation where the various interaction vary in a random way. Our goal is to understand the large time behaviour of the system S under the influence of these random repeated interactions.
This is joint work with Alain Joye and Marco Merkli
Alain Joye (Grenoble)

"Leaky Repeated Interaction Quantum Systems " Slides
We consider a small reference system S interacting with two large quantum systems of a different nature. On the one hand the system S interacts for a fixed duration with the successive elements E of an infinite chain C of identical independentquantum subsystems E. And, on the other hand, it interacts continuously with a heat reservoir R at a given inverse temperature given by an infinitely extended Fermi gas. The reservoir and the chain are not coupled. When the reservoir is absent, the state of the repeated interaction quantum system defined by S and the chain C approches a non-equilibrium asymptotic state for large times. When the chain is absent, the system S and the reservoir R reach an equilibrium state at large times. Our goal is to describe the large time behaviour of the fully coupled system S+R+C and to describe the asymptotic state of this system in the large times limit.

This is joint work with Laurent Bruneau and Marco Merkli
Claude Alain Pillet (Marseille)

"Spectral analysis of a CP map and thermal relaxation of a QED cavity" Slides
I will present the results of a recent joint work with L. Bruneau (Cergy) concerning the thermal relaxation of a QED cavity under repeated interactions with atoms (a so called one-atom maser). The main points of this work are: (i) to study return to equilibrium for some open system with infinite dimensional Hilbert space (results in this direction are scarce); (ii) to show that, using an appropriate version of Perron-Froebenius theory (due to Schrader), one can obtain interesting information on the spectrum of completely positive (CP) maps which can be turned into ergodic properties of the semigroup generated by this map.

Semi-classical non-selfadjoint operators

Mikael Hitrik (UCLA)

"Semiclassical estimates for non-selfadjoint operators with double characteristics" Slides
In this talk, we would like to report on a work in progress together with Karel Pravda-Starov. For a class of non-selfadjoint pseudodifferential operators with double characteristics, we study semiclassical resolvent bounds and asymptotics for the low lying eigenvalues. Specifically, assuming that the quadratic approximations along the characteristics enjoy an ellipticity property along a suitable symplectic subspace of the phase space, we establish semiclassical hypoelliptic a priori estimates with a loss of a full power of the semiclassical parameter. We also investigate the more precise asymptotic description of the low lying eigenvalues as well as semiclassical subelliptic estimates for the operators in question.
Johannes Sjostrand (Paris, Polytechnique)

"Weyl asymptotics for non-selfadjoint differential operators with multiplicative random perturbations". Slides
In this talk we consider non-self-adjoint differential operators with a multiplicative random perturbation either in the semi-classical limit or in the high energy limit. In the first case the eigenvalues distribute according to the Weyl law with probability close to 1. In the second case we have the Weyl distribution of the eigenvalues almost surely. Earlier related results were obtained by M. Hager (2005) and W. Bordeaux Montrieux (2008) in one dimension and by Hager and the speaker in higher dimension. Multiplicative perturbations in higher dimension required quite essential improvements of the methods. These results were obtained in recent works by the speaker and by Bordeaux Montrieux and the speaker. One of the motivations is to obtain similar results for resonances.

San Vu Ngoc (Rennes)

"Complex eigenvalues and KAM quasimodes near rational tori"
Joint work in progress with M. Hitrik. The context is the determination of the spectrum of small non-selfadjoint perturbations of selfadjoint operators. In case the unperturbed hamiltonian has a rational invariant torus, we show that by perturbing it we obtain new families of diophantine KAM tori for which we can apply the non-selfadjoint Bohr-Sommerfeld rules of Melin and Sjöstrand.

Semiclassical Quantum resonances

Tanya Christiansen (Missouri)

"Lower bounds on the order of growth of the resonance counting function in Euclidean scattering"
For certain classes of perturbations of the odd-dimensional Euclidean Laplacian, we show that generically the resonance counting function has maximal order of growth. The classes include Schr\"odinger operators with compactly supported potentials and compactly supported metric perturbations of $\Real^d$. Similar results are true for potential scattering in even dimensions. The proofs use some techniques from the theory of several complex variables. Some of these results are joint work with P. Hislop.
Vesselin Petkov (Bordeaux)

"Estimates of the scattering operator in resonances free domains"
We study the scattering operator $S(z): L^2(S^{n-1}) \longrightarrow L^2(S^{n-1})$ for compact trapping perturbations assuming that has a holomorphic prolongation from $\{z \in {\mathbb{C}}: \mathop{\rm Im}\nolimits z \leq 0\}$ to $U_{\delta} = \{z \in {\mathbb{C}}: 0 < \mathop{\rm Im}\nolimits z \leq \delta\}.$ We will present several results obtained in collaboration with A. Borichev concerning the estimates of $\frac{1}{f(z)}$ in $U_{\delta}$ for functions holomorphic in ${\mathbb{C}}_{+}$ and such that $f(z) \neq 0$ for $z \in U_{\delta}$ . We show that the estimates of $\frac{1}{\vert f(z)\vert}$ depend on the distribution of zeros of lying in $\{z \in {\mathbb{C}}:\:0 <\delta \leq \mathop{\rm Im}\nolimits z \leq k\vert\mathop{\rm Re}\nolimits z\vert,\: k > 0\}.$
Thierry Ramond (Orsay)

"Spectral projection for barrier-top resonances and applications " Slides
We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. Thanks to a resolvent estimate for complex energies close to such a resonance, and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h, and we compute its leading term.
We also give the residu of the scattering amplitude at such a resonance, and as an application, give an expansion of the Schrödinger group in terms of these resonances.
These results have been obtained in collaboration with J.-F. Bony, S. Fujiie and M. Zerzeri.
Maciej Zworski (Berkeley)

"Quantum resonances for classical chaotic systems"
(joint work with S. Nonnenmacher and J. Sjöstrand)

Ruelle Resonances

S. Gouëzel (Rennes)

"Spectral gap for piecewise hyperbolic maps "
In the last few years, several approaches have been devised to study the spectral properties of smooth hyperbolic maps. Unfortunately, none of them could possibly apply to billiards. As a first step in this direction, we have developed with V. Baladi a new framework to deal with hyperbolic maps with discontinuities, combining analytic (anisotropic spaces of Triebel type) and geometric (a class of foliations that are locally invariant under the map) tools. This approach covers a large class of maps but requires a bunching condition, which is probably only an artefact of the method but is always satisfied when the stable or unstable direction is one-dimensional.
O. Bandtlow (Queen Mary, London)

"Asymptotics of Ruelle resonances"
Ruelle resonances are the complex exponential decay-rates of time correlation functions of a dynamical system. If the system is given by a real analytic expanding map and the observables are smooth then it is known that the Ruelle resonances form a sequence converging to zero. It is also known that in this case the Ruelle resonances correspond to the eigenvalues λ_n of a certain compact operator (known as transfer operator in this context), which acts on spaces of holomorphic functions.
The talk will be concerned with the problem of determining the asymptotics of the eigenvalue sequence {λ_n}. In particular, I will discuss a method that yields simple, explicitly computable upper bounds for each λ_n.
This is joint work with O. Jenkinson.
Hans Henrik Rugh (Cergy)

"Resonances, intermittency and Möbius transformations"
Möbius transformations mapping the interval [0,1] into itself have the advantage of admitting nice analytic extensions to the Riemann Sphere. Pertinent examples are the Farey map and the Gauss map.
Associated with these transformations we have Ruelle transfer operators and their Fredholm determinants, and the latter are poised for interesting and surprising analytic properties. We will discuss results for such maps and some related open problems.
Nicolas Roy (Berlin)

"A microlocal approach to Ruelle resonances of Anosov diffeomorphisms"
If f is an Anosov diffeomorphism on a compact manifold, the decay of the dynamical correlation functions is governed by the so-called Ruelle resonances. It follows from the works of Baladi & al and Liverani & al, that these resonances can be obtained by a suitable spectral analysis of the composition operator (or another one related to it) called the "Transfer operator".
In this talk, we will show how these results can be obtained by a systematic microlocal analysis. We will explain also why microlocal analysis is natural in this problem, and mention possible perspectives of the approach.
The talk is based on the recent paper :
[Faure, Roy, Sjoestrand] "Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances". Open Math. Journal (2008), vol. 1, 35--81.

Hyperbolic Manifolds

David Borthwick (Emory)

"Upper and lower bounds on resonances for asymptotically hyperbolic manifolds "
We will present some recent results on bounds on resonances and scattering poles for conformally compact manifolds which are hyperbolic near infinity. Optimal global upper bounds are obtained by applying standard singular value estimates to the resolvent and scattering matrix. These upper bounds lead to a Poisson formula expressing the regularized wave trace as a sum over resonances, which in turn implies an optimal lower bound on the counting functions.
J. Hilgert (Paderborn).

"Resonances and residue operators for symmetric spaces of rank one"

Let X=G/K be a rank one Riemmanian symmetric space of non-compact type and Delta the Laplace-Beltrami operator on X. We show that the resolvent operator R(z) can be meromorphically continued and explcitly determine the poles. Further we describe the residue operators in terms of finite dimensional spherical representations of G. The result answers a question posed by M. Zworski. This is joint work with A. Pasquale (Metz).
D. Mayer (Clausthal)

"Character deformations and the zeros of Selbergs zeta function for \Gamma_0(4) via the transfer operator" Slides Pictures
I will discuss numerical results for the behaviour of the zeros of Selberg's zeta function for the Hecke congruence subgroup \Gamma_0(4) under the singular deformation of a certain character. This character was introduced by A. Selberg when he discussed the distribution of poles of Eisenstein series in the complex plane. Our results are derived from the transfer operator for this group with character.
This is a work done with Markus Fraczek.
Y. Petridis (UCL)

Posters

Olivier Lablée (Grenoble)

"Semi classical spectrum of the double wells Schrödinger operator near the local maximun"
Steffen Loeck (Dresden).

"Dynamical tunneling in systems with a mixed phase space".
The work has been done together with Arnd Baecker and Roland Ketzmerick.
Alexander Potzuweit (Marburg)

"Fractal Weyl law in experimental microwave billiards"
Françoise Truc (Grenoble)

"Magnetic bottles in Euclidian and hyperbolic geometry"