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spectrum and zeta functions in quantum and classical chaos

Résonances et orbites périodiques:

spectre et fonctions zêta du chaos quantique et classique (Ecole thématique CNRS)

June 27-July 5, 2005

ANANTHARAMAN: "Patterson-Sullivan distributions, Ruelle
resonances and quantum
ergodicity in constant negative curvature"

BAECKER: "Flooding of regular islands by chaotic states"

The structure of wave functions in quantum systems strongly depends on the underlying classical dynamics. We illustrate this with several examples for regular and fully chaotic systems. In systems with mixed phase space, where regular and chaotic motion coexist, the wave functions are expected to localize on the corresponding regions in phase space. Surprisingly, this picture is too simple in general. We show that it is possible that chaotic states extend partially and even fully into the region of the regular island.

The structure of wave functions in quantum systems strongly depends on the underlying classical dynamics. We illustrate this with several examples for regular and fully chaotic systems. In systems with mixed phase space, where regular and chaotic motion coexist, the wave functions are expected to localize on the corresponding regions in phase space. Surprisingly, this picture is too simple in general. We show that it is possible that chaotic states extend partially and even fully into the region of the regular island.

BISMUT: "The hypoelliptic Laplacian"

CVITANOVIC:
"Hopf's last hope:
Spatiotemporal chaos in terms of unstable recurrent patterns"

If everything in a turbulent systems is in constant flux, how is it that humans are able to distinguish different kinds of turbulence? Hopf's answer was that dynamics drives a given spatially extended system through a repertoire of unstable patterns; we identify a particular kind of turbulence by catching every so often a glimpse of a familiar pattern. Recent experimental and theoretical advances in observation and computation of unstable coherent structures in turbulence close to its onset support this view. A predictive dynamical theory of turbulence for moderate Rayleigh number full Navier-Stokes flows is within reach. I will illustrate Hopf's program by an investigation of a 1-dimensional spatially extended Kuramoto-Sivashinsky system, a PDE that describes interfacial instabilities such as unstable flame fronts.

If everything in a turbulent systems is in constant flux, how is it that humans are able to distinguish different kinds of turbulence? Hopf's answer was that dynamics drives a given spatially extended system through a repertoire of unstable patterns; we identify a particular kind of turbulence by catching every so often a glimpse of a familiar pattern. Recent experimental and theoretical advances in observation and computation of unstable coherent structures in turbulence close to its onset support this view. A predictive dynamical theory of turbulence for moderate Rayleigh number full Navier-Stokes flows is within reach. I will illustrate Hopf's program by an investigation of a 1-dimensional spatially extended Kuramoto-Sivashinsky system, a PDE that describes interfacial instabilities such as unstable flame fronts.

DETTMANN:
"Open circular billiards and the Riemann
hypothesis"

A comparison of escape rates from one and from two holes in an experimental container (e.g., a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeros of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis.

A comparison of escape rates from one and from two holes in an experimental container (e.g., a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeros of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis.

GREMAUD:
"Semi-classical analysis of real
atomic spectra beyond Gutzwiller's approximation"

The validity of semiclassical expansions in the power of $\hbar$ for the quantum Green's function have been extensively tested for billiards systems, but not in the case of smooth potential, like for atomic systems (hydrogen atom in a magnetic field, helium atom...). In this talk, I will present an efficient method, based on the classical Green's functions, allowing the calculation of \hbar corrections for the propagator, the quantum Green's function, and their traces. In addition, I will explain how discrete symmetries and centrifugal terms must be taken into account in order to obtain the proper \hbar expansion of semiclassical expressions in the case of the hydrogen atom in a magnetic field.

The validity of semiclassical expansions in the power of $\hbar$ for the quantum Green's function have been extensively tested for billiards systems, but not in the case of smooth potential, like for atomic systems (hydrogen atom in a magnetic field, helium atom...). In this talk, I will present an efficient method, based on the classical Green's functions, allowing the calculation of \hbar corrections for the propagator, the quantum Green's function, and their traces. In addition, I will explain how discrete symmetries and centrifugal terms must be taken into account in order to obtain the proper \hbar expansion of semiclassical expressions in the case of the hydrogen atom in a magnetic field.

JAKOBSON: "Estimates from below for the remainder in local Weyl's law"
(with Iosif Polterovich)

We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl's law on compact manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Our results can be considered pointwise versions (on a general manifold) of Hardy's lower bounds for the error term in the Gauss circle problem.

See http://front.math.ucdavis.edu/math.SP/0505400

Slides available here

We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl's law on compact manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Our results can be considered pointwise versions (on a general manifold) of Hardy's lower bounds for the error term in the Gauss circle problem.

See http://front.math.ucdavis.edu/math.SP/0505400

Slides available here

KURLBERG:
"On the distribution of matrix elements for the quantum cat map"

For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable. We will study the fluctuations of the matrix elements for the desymmetrized quantum cat map and present a conjecture for the distribution of the normalized matrix elements, namely that their distribution is that of a certain weighted sum of traces of independent matrices in SU(2). This is in contrast to generic chaotic systems where the distribution is expected to be Gaussian. We will show that the second and fourth moment of the distribution agree with the conjecture, and also present some numerical evidence.

For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable. We will study the fluctuations of the matrix elements for the desymmetrized quantum cat map and present a conjecture for the distribution of the normalized matrix elements, namely that their distribution is that of a certain weighted sum of traces of independent matrices in SU(2). This is in contrast to generic chaotic systems where the distribution is expected to be Gaussian. We will show that the second and fourth moment of the distribution agree with the conjecture, and also present some numerical evidence.

LAKSHMINARAYAN: "Multifractal eigenstates of quantum chaos
and
the Thue-Morse sequence"

We show the emergence of automatic sequences such as the Thue-Morse sequence and some of its generalizations in the eigenstates of the quantum baker's map, a simple paradigmatic model of quantizing chaos. This enables us to write nearly exactly a class of states in terms of the spectra of such sequences, which turn out to be multifractal measures. These states include those that are strongly scarred by short classical periodic orbits, and their homoclinic excursions. Thus the deterministic structural disorder of these quantum chaotic states is seen to be captured by these ubiquitous, simple yet intriguing, sequences. While this happens for Hilbert space dimensions that are powers of two, we touch upon the more general case where using an exactly solvable permutation operator we "simplify" states of the quantum baker's map. We point out that this operator is a crucial component of both the art of shuffling cards, as well as the celebrated quantum algorithm of Shor that factorizes large numbers with ease.

We discuss the Selberg zeta function associated to the lift of the geodesic flow to a vector bundle on the unit sphere bundle of an infinite volume hyperbolic manifold without cusps. We are particularly interested in the understanding of its zeroes and poles. Usually, one tries to establish a Selberg type trace formula which leads to a description in terms of resonances and topological data. Inspired by a conjecture of Patterson, we propose an alternative description in terms of distributions supported on the limit set which are covariant with respect to the action of the fundamental group. At least philosophically, this description is very close to the transfer operator approach. In order to get correct formulas for all singularities, one has to consider not only covariant distributions on the limit set, but also higher cohomology classes with coefficients in the space of distributions supported on the limit set.

We show the emergence of automatic sequences such as the Thue-Morse sequence and some of its generalizations in the eigenstates of the quantum baker's map, a simple paradigmatic model of quantizing chaos. This enables us to write nearly exactly a class of states in terms of the spectra of such sequences, which turn out to be multifractal measures. These states include those that are strongly scarred by short classical periodic orbits, and their homoclinic excursions. Thus the deterministic structural disorder of these quantum chaotic states is seen to be captured by these ubiquitous, simple yet intriguing, sequences. While this happens for Hilbert space dimensions that are powers of two, we touch upon the more general case where using an exactly solvable permutation operator we "simplify" states of the quantum baker's map. We point out that this operator is a crucial component of both the art of shuffling cards, as well as the celebrated quantum algorithm of Shor that factorizes large numbers with ease.

(1) "Multifractal eigenstates of quantum chaos and the Thue-Morse
sequence",
N. Meenakshisundaram, and AL. (nlin.CD/0412042, to
appear in the Phys. Rev. E rapid comm..)

(2) "Shuffling cards, factoring numbers, and the quantum baker's map", AL. (nlin.CD/0505057).

(2) "Shuffling cards, factoring numbers, and the quantum baker's map", AL. (nlin.CD/0505057).

LEBOEUF:
"Thermodynamics of a Fermi gas: average, fluctuations, universality"

The thermodynamic properties of Fermi gases show, on top of an average behavior, characteristic fluctuations (as a function, for instance, of the number of particles in the gas). These fluctuations may be described, in a semiclassical theory, as sums over the periodic orbits of the corresponding classical potential. We present a general analysis of these fluctuations and describe how they depend on the regular or chaotic nature of the classical dynamics. The typical size and correlations of the fluctuations may be expressed, generically, in terms of Ruelle--Pollicott resonances. The calculation of the level density of the gas is of particular interest. We show how this quantity is related to the problem of computing the number of partitions of an integer, and discuss the natural extensions (some of them connected to random matrix theory) that emerge.

The thermodynamic properties of Fermi gases show, on top of an average behavior, characteristic fluctuations (as a function, for instance, of the number of particles in the gas). These fluctuations may be described, in a semiclassical theory, as sums over the periodic orbits of the corresponding classical potential. We present a general analysis of these fluctuations and describe how they depend on the regular or chaotic nature of the classical dynamics. The typical size and correlations of the fluctuations may be expressed, generically, in terms of Ruelle--Pollicott resonances. The calculation of the level density of the gas is of particular interest. We show how this quantity is related to the problem of computing the number of partitions of an integer, and discuss the natural extensions (some of them connected to random matrix theory) that emerge.

MAYER:
"Transfer operators and the automorphic functions for
modular groups"

I will review in this talk the relation between certain eigenfunctions of the transfer operators for the geodesic flows on modular surfaces, the so called period functions, and the holomorphic and real analytic automorphic functions for the corresponding modular groups.

I will review in this talk the relation between certain eigenfunctions of the transfer operators for the geodesic flows on modular surfaces, the so called period functions, and the holomorphic and real analytic automorphic functions for the corresponding modular groups.

NONNENMACHER:
"Open quantum bakers" (joint with M.Zworski)

We analyze the quantum version of a simple chaotic map on the torus (the generalized baker's map), from the semiclassical point of view. This system should serve as a model for more realistic systems of quantum chaotic scattering (e.g. a 3-disk scatterer on the plane). The classical map has a fractal invariant hyperbolic set ("repeller"), and invariant measures are easy to construct explicitly. We are interested in the spectrum of the quantum "propagator", which should be identified with the resonances in the scattering framework. A first version of the quantized map provides a numerical confirmation of the fractal Weyl law for the density of quantum resonances (cf. M.Zworski's course). The exponent in that law is related to the dimension of the classical repeller. We also discuss the "profile function" (prefactor) of the density, and the eigenstates associated with the resonances. Within a simplified quantization scheme, we manage to analytically compute the full resonance spectrum: the latter is shown to satisfy the fractal Weyl law. Some of the eigenstates have a simple fractal structure in phase space.

We analyze the quantum version of a simple chaotic map on the torus (the generalized baker's map), from the semiclassical point of view. This system should serve as a model for more realistic systems of quantum chaotic scattering (e.g. a 3-disk scatterer on the plane). The classical map has a fractal invariant hyperbolic set ("repeller"), and invariant measures are easy to construct explicitly. We are interested in the spectrum of the quantum "propagator", which should be identified with the resonances in the scattering framework. A first version of the quantized map provides a numerical confirmation of the fractal Weyl law for the density of quantum resonances (cf. M.Zworski's course). The exponent in that law is related to the dimension of the classical repeller. We also discuss the "profile function" (prefactor) of the density, and the eigenstates associated with the resonances. Within a simplified quantization scheme, we manage to analytically compute the full resonance spectrum: the latter is shown to satisfy the fractal Weyl law. Some of the eigenstates have a simple fractal structure in phase space.

OLBRICH:
"Group cohomology and the Selberg zeta function"

(joint with U. Bunke)
We discuss the Selberg zeta function associated to the lift of the geodesic flow to a vector bundle on the unit sphere bundle of an infinite volume hyperbolic manifold without cusps. We are particularly interested in the understanding of its zeroes and poles. Usually, one tries to establish a Selberg type trace formula which leads to a description in terms of resonances and topological data. Inspired by a conjecture of Patterson, we propose an alternative description in terms of distributions supported on the limit set which are covariant with respect to the action of the fundamental group. At least philosophically, this description is very close to the transfer operator approach. In order to get correct formulas for all singularities, one has to consider not only covariant distributions on the limit set, but also higher cohomology classes with coefficients in the space of distributions supported on the limit set.

POLLICOTT: "Uniqueness of g-measures"
(joint with Anders Johansson and Anders Oberg)

g-measures are invariant probability measures m for a (one sided) subshift of finite type T:X -> X for which the Jacobian 1/g := d(mT)/dm is continuous. These were introduced into ergodic theory by Keane. We consider the classical problem of which functions g have a unique m satisfying this identity (i.e., a unique g-measure). In particular, we present a new approach to showing uniqueness under the square summability condition of Johansson and Oberg which allows us extend the results to a slightly broader setting.

g-measures are invariant probability measures m for a (one sided) subshift of finite type T:X -> X for which the Jacobian 1/g := d(mT)/dm is continuous. These were introduced into ergodic theory by Keane. We consider the classical problem of which functions g have a unique m satisfying this identity (i.e., a unique g-measure). In particular, we present a new approach to showing uniqueness under the square summability condition of Johansson and Oberg which allows us extend the results to a slightly broader setting.

RUDNICK: "Fluctuations of matrix elements and moments of zeta functions" (joint with K. Soundararajan)

The moments of the Riemann zeta function on the critical line have been an object of study for several decades. Analogously one can study the moments of central values of families of L-functions. Using results from Random Matrix Theory, Keating and Snaith have recently given precise conjectures for the limiting values of moments of central values of various families of L-functions. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of L-functions. An application of these results is to a problem in "quantum chaos": For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable ("Quantum Unique Ergodicity"). One then wants to describe the fluctuations of the matrix elements around the phase space average. For generic systems, it has been conjectured by Feingold and Peres that the matrix coefficients have a limiting variance which decays at a certain rate with a pre-factor which is given by the classical variance of the observable, and that the value distribution is Gaussian. We study this question for the modular domain. In this case, for certain observables, the square of the matrix coefficients is essentially the central value of an L-function (Watson's formula). Luo and Sarnak have computed the variance and found that while the rate of decay matches the Feingold-Peres predictions, the pre-factor differs from the classical variance by an arithmetic quantity. we study the the higher moments of the normalized matrix coefficients and show that they blow up, therefore the fluctuations in this case are not Gaussian.

The moments of the Riemann zeta function on the critical line have been an object of study for several decades. Analogously one can study the moments of central values of families of L-functions. Using results from Random Matrix Theory, Keating and Snaith have recently given precise conjectures for the limiting values of moments of central values of various families of L-functions. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of L-functions. An application of these results is to a problem in "quantum chaos": For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable ("Quantum Unique Ergodicity"). One then wants to describe the fluctuations of the matrix elements around the phase space average. For generic systems, it has been conjectured by Feingold and Peres that the matrix coefficients have a limiting variance which decays at a certain rate with a pre-factor which is given by the classical variance of the observable, and that the value distribution is Gaussian. We study this question for the modular domain. In this case, for certain observables, the square of the matrix coefficients is essentially the central value of an L-function (Watson's formula). Luo and Sarnak have computed the variance and found that while the rate of decay matches the Feingold-Peres predictions, the pre-factor differs from the classical variance by an arithmetic quantity. we study the the higher moments of the normalized matrix coefficients and show that they blow up, therefore the fluctuations in this case are not Gaussian.

ZAGIER: "Automorphic forms and their period functions"

ZELDITCH: "Complex zeros of real ergodic eigenfunctions"

Little is known about the distribution of real zeros of real valued eigenfunctions of the Laplacian on a compact analytic Riemannian manifold (M, g). If Delta phi_{\lambda} = \lambda^2 \phi_{\lambda}, then the best result (Donnelly-Fefferman) is that the (n-1) dimensional measure of Z_{\lambda} = {phi_{\lambda} = 0} is bounded above and below by a constant times lambda. In our talk, we simplify the problem by complexifying it. We holomorphically continue phi_{\lambda} to the cotangent bundle (equipped with its adapted complex structure) and study the limit distribution of its complex zeros. In the case where {\phi_{\lambda}} is a sequence of ergodic eigefunctions, the limit distribution of complex zeros is given by the (1,1) form ddbar |\xi|_g. In particular, if the geodesic flow of (M,g) is ergodic, this limit formula is valid for 100% of the eigenfunctions.

Little is known about the distribution of real zeros of real valued eigenfunctions of the Laplacian on a compact analytic Riemannian manifold (M, g). If Delta phi_{\lambda} = \lambda^2 \phi_{\lambda}, then the best result (Donnelly-Fefferman) is that the (n-1) dimensional measure of Z_{\lambda} = {phi_{\lambda} = 0} is bounded above and below by a constant times lambda. In our talk, we simplify the problem by complexifying it. We holomorphically continue phi_{\lambda} to the cotangent bundle (equipped with its adapted complex structure) and study the limit distribution of its complex zeros. In the case where {\phi_{\lambda}} is a sequence of ergodic eigefunctions, the limit distribution of complex zeros is given by the (1,1) form ddbar |\xi|_g. In particular, if the geodesic flow of (M,g) is ergodic, this limit formula is valid for 100% of the eigenfunctions.

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