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 Annales Henri Poincaré   [SJR: 1.016]   [H-I: 28]   [3 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1424-0637 - ISSN (Online) 1424-0661    Published by Springer-Verlag  [2302 journals]
• Structural Stability of a Dynamical System Near a Non-Hyperbolic Fixed
Point
• Abstract: Abstract We prove structural stability under perturbations for a class of discrete-time dynamical systems near a non-hyperbolic fixed point. We reformulate the stability problem in terms of the well-posedness of an infinite-dimensional nonlinear ordinary differential equation in a Banach space of carefully weighted sequences. Using this, we prove existence and regularity of flows of the dynamical system which obey mixed initial and final boundary conditions. The class of dynamical systems we study, and the boundary conditions we impose, arise in a renormalization group analysis of the 4-dimensional weakly self-avoiding walk and the 4-dimensional n-component φ 4 spin model.
PubDate: 2015-04-01

• The Dynamics of a Class of Quasi-Periodic Schrödinger Cocycles
• Abstract: Abstract Let $${f:\mathbb{T} \to\mathbb{R}}$$ be a Morse function of class C 2 with exactly two critical points, let $${\omega \in \mathbb{T}}$$ be Diophantine, and let λ > 0 be sufficiently large (depending on f and ω). For any value of the parameter $${E\in \mathbb{R}}$$ , we make a careful analysis of the dynamics of the skew-product map $$\Phi_E(\theta,r)=\left(\theta+\omega, {\rm \lambda} f(\theta)-E-1/r\right),$$ acting on the “torus” $${\mathbb{T} \times \widehat{\mathbb{R}}}$$ . Here, $${\widehat{\mathbb{R}}}$$ denotes the projective space $${\mathbb{R} \cup\{\infty\}}$$ . The map Φ E is intimately related to the quasi-periodic Schrödinger cocycle $${(\omega, A_E): \mathbb{T}\times \mathbb{R}^2 \to \mathbb{T}\times \mathbb{R}^2,\, (\theta,x)\mapsto (\theta+\omega, A_E(\theta)\cdot x)}$$ , where $${A_E:\mathbb{T}\to {\rm SL}(2,\mathbb{R})}$$ is given by $$A_{E}(\theta)=\left( \begin{array}{ll}0 \quad \quad \quad 1\\ -1 \quad {\rm \lambda} f(\theta)-E \\\end{array}\right),\quad E \in \mathbb{R}.$$ More precisely, (ω, A E ) naturally acts on the space $${\mathbb{T} \times \widehat{\mathbb{R}}}$$ , and Φ E is the map thus obtained. The cocycle (ω, A E ) arises when investigating the eigenvalue equation H θ u = Eu, where H θ is the quasi-periodic Schrödinger operator $$(H_\theta u)_n=-(u_{n+1}+u_{n-1}) + {\rm \lambda} f(\theta+(n-1)\omega)u_n,$$ acting on the space $${l^2(\mathbb{Z})}$$ . It is well known that the spectrum of $${H_\theta,\, \sigma(H)}$$ , is independent of the phase $${\theta \in \mathbb{T}}$$ . Under our assumptions on f, ω and λ, Sinai (in J Stat Phys 46(5–6):861–909, 1987) has shown that σ(H) is a Cantor set, and the operator H θ has a pure-point spectrum, with exponentially decaying eigenfunctions, for a.e. $${\theta \in \mathbb{T}}$$ The analysis of Φ E allows us to derive three main results: The (maximal) Lyapunov exponent of the Schrödinger cocycle (ω, A E ) is $${\gtrsim {\rm log} {\rm \lambda}}$$ , uniformly in
PubDate: 2015-04-01

• Approximation of the Integrated Density of States on Sofic Groups
• Abstract: Abstract In this paper, we study spectral properties of self-adjoint operators on a large class of geometries given via sofic groups. We prove that the associated integrated densities of states can be approximated via finite volume analogues. This is investigated in the deterministic as well as in the random setting. In both cases, we cover a wide range of operators including in particular unbounded ones. The large generality of our setting allows one to treat applications from long-range percolation and the Anderson model. Our results apply to operators on $${\mathbb{Z}^d}$$ , amenable groups, residually finite groups and therefore in particular to operators on trees. All convergence results are established without an ergodic theorem at hand.
PubDate: 2015-04-01

• An Extensive Adiabatic Invariant for the Klein–Gordon Model in the
Thermodynamic Limit
• Abstract: Abstract We construct an extensive adiabatic invariant for a Klein–Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant a, the evolution of the adiabatic invariant is controlled up to time scaling as β 1/a for any large enough value of the inverse temperature β. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.
PubDate: 2015-04-01

• Eisenstein Quasimodes and QUE
• Abstract: Abstract We consider the question of quantum unique ergodicity (QUE) for quasimodes on surfaces of constant negative curvature, and conjecture the order of quasimodes that should satisfy QUE. We then show that this conjecture holds for Eisenstein series on $${SL(2,\mathbb{Z})\backslash\mathbb{H}}$$ , extending results of Luo–Sarnak and Jakobson. Moreover, we observe that the equidistribution results of Luo–Sarnak and Jakobson extend to quasimodes of much weaker order—for which QUE is known to fail on compact surfaces—though in this scenario the total mass of the limit measures will decrease. We interpret this stronger equidistribution property in the context of arithmetic QUE, in light of recent joint work with Lindenstrauss (Invent Math 198(1), 219–259, 2014) on joint quasimodes.
PubDate: 2015-03-19

• Optimal Lower Bound of the Resonance Widths for the Helmholtz Resonator
• Abstract: Abstract Under a geometric assumption on the region near the end of its neck, we prove an optimal exponential lower bound on the widths of resonances for a general two-dimensional Helmholtz resonator. An extension of the result to the n-dimensional case, $${n \leq 12}$$ , is also obtained.
PubDate: 2015-03-17

• Dynamical Locality of the Free Maxwell Field
• Abstract: Abstract The extent to which the non-interacting and source-free Maxwell field obeys the condition of dynamical locality is determined in various formulations. Starting from contractible globally hyperbolic spacetimes, we extend the classical field theory to globally hyperbolic spacetimes of arbitrary topology in two ways, obtaining a ‘universal’ theory and a ‘reduced’ theory of the classical free Maxwell field and their corresponding quantisations. We show that the classical and the quantised universal theory fail local covariance and dynamical locality owing to the possibility of having non-trivial radicals in the classical pre-symplectic spaces and non-trivial centres in the quantised *-algebras. The classical and the quantised reduced theory are both locally covariant and dynamically local, thus closing a gap in the discussion of dynamical locality and providing new examples relevant to the question of how theories should be formulated so as to describe the same physics in all spacetimes.
PubDate: 2015-03-06

• The Altshuler–Shklovskii Formulas for Random Band Matrices II: The
General Case
• Abstract: Abstract The Altshuler–Shklovskii formulas (Altshuler and Shklovskii, BZh Eksp Teor Fiz 91:200, 1986) predict, for any disordered quantum system in the diffusive regime, a universal power law behaviour for the correlation functions of the mesoscopic eigenvalue density. In this paper and its companion (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013), we prove these formulas for random band matrices. In (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013) we introduced a diagrammatic approach and presented robust estimates on general diagrams under certain simplifying assumptions. In this paper, we remove these assumptions by giving a general estimate of the subleading diagrams. We also give a precise analysis of the leading diagrams which give rise to the Altschuler–Shklovskii power laws. Moreover, we introduce a family of general random band matrices which interpolates between real symmetric (β = 1) and complex Hermitian (β = 2) models, and track the transition for the mesoscopic density–density correlation. Finally, we address the higher-order correlation functions by proving that they behave asymptotically according to a Gaussian process whose covariance is given by the Altshuler–Shklovskii formulas.
PubDate: 2015-03-01

• Neutrino Radiation Showing a Christodoulou Memory Effect in General
Relativity
• Abstract: Abstract We describe neutrino radiation in general relativity by introducing the energy–momentum tensor of a null fluid into the Einstein equations. Investigating the geometry and analysis at null infinity, we prove that a component of the null fluid enlarges the Christodoulou memory effect of gravitational-waves. The description of neutrinos in general relativity as a null fluid can be regarded as a limiting case of a more general description using the massless limit of the Einstein–Vlasov system. Gigantic neutrino bursts occur in our universe in core-collapse supernovae and in the mergers of neutron star binaries.
PubDate: 2015-03-01

• Deformations of Axially Symmetric Initial Data and the Mass-Angular
Momentum Inequality
• Abstract: Abstract We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality (in a weak sense) for the scalar curvature; this answers a question posed by R. Schoen. The primary equation involved, bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Each equation in the system is analyzed in detail individually, and it is shown that appropriate existence/uniqueness results hold with the solution satisfying desired asymptotics. Lastly, it is shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass and angular momentum.
PubDate: 2015-03-01

• Multi-Species Mean Field Spin Glasses. Rigorous Results
• Abstract: Abstract We study a multi-species spin glass system where the density of each species is kept fixed at increasing volumes. The model reduces to the Sherrington–Kirkpatrick one for the single species case. The existence of the thermodynamic limit is proved for all density values under a convexity condition on the interaction. The thermodynamic properties of the model are investigated and the annealed, the replica-symmetric and the replica symmetry breaking bounds are proved using Guerra’s scheme. The annealed approximation is proved to be exact under a high-temperature condition. We show that the replica-symmetric solution has negative entropy at low temperatures. We study the properties of a suitably defined replica symmetry breaking solution and we optimize it within a novel ziggurat ansatz. The generalized order parameter is described by a Parisi-like partial differential equation.
PubDate: 2015-03-01

• Construction of Real-Valued Localized Composite Wannier Functions for
Insulators
• Abstract: Abstract We consider a real periodic Schrödinger operator and a physically relevant family of $${m \geq 1}$$ Bloch bands, separated by a gap from the rest of the spectrum, and we investigate the localization properties of the corresponding composite Wannier functions. To this aim, we show that in dimension $${d \leq 3}$$ , there exists a global frame consisting of smooth quasi-Bloch functions which are both periodic and time-reversal symmetric. Aiming to applications in computational physics, we provide a constructive algorithm to obtain such a Bloch frame. The construction yields the existence of a basis of composite Wannier functions which are real-valued and almost-exponentially localized. The proof of the main result exploits only the fundamental symmetries of the projector on the relevant bands, allowing applications, beyond the model specified above, to a broad range of gapped periodic quantum systems with a time-reversal symmetry of bosonic type.
PubDate: 2015-02-28

• Gupta–Bleuler Quantization of the Maxwell Field in Globally
Hyperbolic Space-Times
• Abstract: Abstract We give a complete framework for the Gupta–Bleuler quantization of the free electromagnetic field on globally hyperbolic space-times. We describe one-particle structures that give rise to states satisfying the microlocal spectrum condition. The field algebras in the so-called Gupta–Bleuler representations satisfy the time-slice axiom, and the corresponding vacuum states satisfy the microlocal spectrum condition. We also give an explicit construction of ground states on ultrastatic space-times. Unlike previous constructions, our method does not require a spectral gap or the absence of zero modes. The only requirement, the absence of zero-resonance states, is shown to be stable under compact perturbations of topology and metric. Usual deformation arguments based on the time-slice axiom then lead to a construction of Gupta–Bleuler representations on a large class of globally hyperbolic space-times. As usual, the field algebra is represented on an indefinite inner product space, in which the physical states form a positive semi-definite subspace. Gauge transformations are incorporated in such a way that the field can be coupled perturbatively to a Dirac field. Our approach does not require any topological restrictions on the underlying space-time.
PubDate: 2015-02-27

• Open Quantum Random Walks: Reducibility, Period, Ergodic Properties
• Abstract: Abstract We study the analogues of irreducibility, period, and communicating classes for open quantum random walks, as defined in (J Stat Phys 147(4):832–852, 2012). We recover results similar to the standard ones for Markov chains, in terms of ergodic behaviour, decomposition into irreducible subsystems, and characterization of invariant states.
PubDate: 2015-02-19

• On the Existence of a Maximal Cauchy Development for the Einstein
Equations: a Dezornification
• Abstract: Abstract In 1969, Choquet-Bruhat and Geroch established the existence of a unique maximal globally hyperbolic Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn’s lemma. In this paper, we present a proof that avoids the use of Zorn’s lemma. In particular, we provide an explicit construction of this maximal globally hyperbolic development.
PubDate: 2015-02-15

• Semi-Classical Analysis of Non-Self-Adjoint Transfer Matrices in
Statistical Mechanics I
• Abstract: Abstract We propose a way to study one-dimensional statistical mechanics models with complex-valued action using transfer operators. The argument consists of two steps. First, the contour of integration is deformed so that the associated transfer operator is a perturbation of a normal one. Then the transfer operator is studied using methods of semi-classical analysis. In this paper, we concentrate on the second step, the main technical result being a semi-classical estimate for powers of an integral operator which is approximately normal.
PubDate: 2015-02-15

• Mixing for Some Non-Uniformly Hyperbolic Systems
• Abstract: In this work, we present an abstract framework that allows to obtain mixing (and in some cases sharp mixing) rates for a reasonable large class of invertible systems preserving an infinite measure. The examples explicitly considered are the invertible analogue of both Markov and non-Markov unit interval maps. For these examples, in addition to optimal results on mixing and rates of mixing in the infinite case, we obtain results on the decay of correlation in the finite case of invertible non-Markov maps, which, to our knowledge, were not previously addressed. The proposed method consists of a combination of the framework of operator renewal theory, as introduced in the context of dynamical systems by Sarig (Invent Math 150:629–653, 2002), with the framework of function spaces of distributions developed in the recent years along the lines of Blank et al. (Nonlinearity 15:1905–1973, 2001).
PubDate: 2015-02-14

• On the Validity of the Definition of Angular Momentum in General
Relativity
• Abstract: Abstract We exam the validity of the definition of the ADM angular momentum without the parity assumption. Explicit examples of asymptotically flat hypersurfaces in the Minkowski spacetime with zero ADM energy–momentum vector and finite non-zero angular momentum vector are presented. We also discuss the Beig–Ó Murchadha–Regge–Teitelboim center of mass and study analogous examples in the Schwarzschild spacetime.
PubDate: 2015-02-11

• Semi-Classical Measures on Quantum Graphs and the Gauß Map of the
Determinant Manifold
• Abstract: Abstract In this paper, we describe the weak limits of the measures associated to the eigenfunctions of the Laplacian on a Quantum graph for a generic metric in terms of the Gauss map of the determinant manifold. We describe also all the limits with minimal support (the “scars”).
PubDate: 2015-02-01

• Transformations de Bargmann Discrètes Attachées aux Niveaux de
Landau sur la Sphère de Riemann
• Abstract: Abstract We construct a family of transforms labeled by (ν, m) and mapping isometrically square integrable functions on a finite subset of $${\mathbb{R}}$$ onto L 2-eigenspaces associated with the discrete spectrum of a charged particle evolving in the Riemann sphere under influence of a uniform magnetic field with a strength proportional to $${2\nu \in \mathbb{Z}_{+}^{\ast}}$$ . These transforms are attached to spherical Landau levels $${\lambda _{m}^{\nu}=\left( 2m+1\right) \nu +m\left( m+1\right)}$$ with $${m\in \mathbb{Z}_{+}}$$ and are called discrete Bargmann transforms.
PubDate: 2015-02-01

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