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 Annales Henri Poincaré   [SJR: 1.016]   [H-I: 28]   [1 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1424-0637 - ISSN (Online) 1424-0661    Published by Springer-Verlag  [2299 journals]
• A Spacetime Characterization of the Kerr-NUT-(A)de Sitter and Related
Metrics
• Abstract: Abstract A characterization of the Kerr-NUT-(A)de Sitter metric among four dimensional Λ-vacuum spacetimes admitting a Killing vector ξ is obtained in terms of the proportionality of the self-dual Weyl tensor and a natural self-dual double two-form constructed from the Killing vector. This result recovers and extends a previous characterization of the Kerr and Kerr-NUT metrics (Mars, Class Quant Grav 16:2507–2523, 1999). The method of proof is based on (i) the presence of a second Killing vector field which is built in terms of geometric information arising from the Killing vector ξ exclusively, and (ii) the existence of an interesting underlying geometric structure involving a Riemannian submersion of a conformally related metric, both of which may be of independent interest. Other related metrics can also be similarly characterized, in particular the Λ < 0 “black branes” recently used in AdS/CFT correspondence to describe, via holography, the physics of Quark–Gluon plasma.
PubDate: 2015-07-01

• A Limit Equation Criterion for Solving the Einstein Constraint Equations
on Manifolds with Ends of Cylindrical Type
• Abstract: Abstract We prove that in a certain class of conformal data on a manifold with ends of cylindrical type, if the conformally decomposed Einstein constraint equations do not admit a solution, then one can always find a nontrivial solution to the limit equation first explored by Dahl et al. (Duke Math J 161(14):2669–2798, 2012). We also give an example of a Ricci curvature condition on the manifold which precludes the existence of a solution to this limit equation. This shows that the limit equation criterion can be a useful tool for proving the existence of solutions to the Einstein constraint equations on manifolds with ends of cylindrical type.
PubDate: 2015-07-01

• Generalizations of Poisson Structures Related to Rational Gaudin Model
• Abstract: Abstract The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra. Motivated by this analogy, we realize a braiding of the mentioned Poisson structure, i.e. we introduce a “braided Poisson” algebra associated with an involutive solution to the quantum Yang–Baxter equation. Also, we exhibit another generalization of the Gaudin type Poisson structure by replacing the first derivative in the current parameter, entering the so-called local form of this structure, by a higher order derivative. Finally, we introduce a structure, which combines both generalizations. Some commutative families in the corresponding braided Poisson algebra are found.
PubDate: 2015-07-01

• Erratum to: Excitation Spectrum of Interacting Bosons in the Mean-Field
Infinite-Volume Limit
• PubDate: 2015-07-01

• Chaotic Dynamics in an Impact Problem
• Abstract: Abstract We consider the model describing the vertical motion of a ball falling with constant acceleration on a wall and elastically reflected. The wall is supposed to move in the vertical direction according to a given periodic function f. We show that a modification of a method of Angenent based on sub- and super-solutions can be applied in order to detect chaotic dynamics. Using the theory of exact symplectic twist maps of the cylinder one can prove the result under “natural” conditions on the function f.
PubDate: 2015-07-01

• Explicit Riemannian Manifolds with Unexpectedly Behaving Center of Mass
• Abstract: The (relativistic) center of mass (CoM) of an asymptotically flat Riemannian manifold is often defined by certain surface integral expressions evaluated along a foliation of the manifold near infinity, e.g. by Arnowitt, Deser, and Misner (ADM). There are also what we call abstract definitions of the CoM in terms of a foliation near infinity itself, going back to the constant mean curvature (CMC-) foliation studied by Huisken and Yau; these give rise to surface integral expressions when equipped with suitable systems of coordinates. We discuss subtle asymptotic convergence issues regarding the ADM- and the coordinate expressions related to the CMC-CoM. In particular, we give explicit examples demonstrating that both can diverge—in a setting where Einstein’s equation is satisfied. We also give explicit examples of the same asymptotic order of decay with prescribed mass and CoM. We illustrate both phenomena by providing analogs in Newtonian gravity. Our examples conflict with some results in the literature.
PubDate: 2015-07-01

• On Perturbations of Extreme Kerr–Newman Black Holes and their
Evolution
• Abstract: Abstract Using black hole inequalities and the increase of the horizon’s areas, we show that there are arbitrarily small electro-vacuum perturbations of the standard initial data of the extreme Reissner–Nordström black hole that (by contradiction) cannot decay in time into any extreme Kerr–Newman black hole. This proves that, in a formal sense, the reduced family of the extreme Kerr–Newman black holes is unstable. It remains of course to be seen whether the whole family of charged black holes, including those extremes, is stable or not.
PubDate: 2015-07-01

• Accurate Semiclassical Spectral Asymptotics for a Two-Dimensional Magnetic
Schrödinger Operator
• Abstract: Abstract We revisit the problem of semiclassical spectral asymptotics for a pure magnetic Schrödinger operator on a two-dimensional Riemannian manifold. We suppose that the minimal value b 0 of the intensity of the magnetic field is strictly positive, and the corresponding minimum is unique and non-degenerate. The purpose is to get the control on the spectrum in an interval $${(hb_0, h(b_0 + \gamma_0)]}$$ for some $${\gamma_0 > 0}$$ independent of the semiclassical parameter h. The previous papers by Helffer–Mohamed and by Helffer–Kordyukov were only treating the ground-state energy or a finite (independent of h) number of eigenvalues. Note also that N. Raymond and S. Vũ Ngọc have recently developed a different approach of the same problem.
PubDate: 2015-07-01

• Chambers’s Formula for the Graphene and the Hou Model with Kagome
Periodicity and Applications
• Abstract: Abstract The aim of this article is to prove that for the graphene model like for a model considered by the physicist Hou on a kagome lattice, there exists a formula which is similar to the one obtained by Chambers for the Harper model in the case of the rational flux. As an application, we propose a semi-classical analysis of the spectrum of the Hou butterfly near a flat band.
PubDate: 2015-06-03

• Characterization of the Quasi-Stationary State of an Impurity Driven by
Monochromatic Light II: Microscopic Foundations
• Abstract: Abstract From quantum mechanical first principles only, we rigorously study the time-evolution of a N-level atom (impurity) interacting with an external monochromatic light source within an infinite system of free electrons at thermal equilibrium (reservoir). In particular, we establish the relation between the full dynamics of the compound system and the effective dynamics for the N-level atom, which is studied in detail in Bru et al. (Ann Henri Poincaré 13(6):1305–1370, 2012). Together with Bru et al. (Ann Henri Poincaré 13(6):1305–1370, 2012) the present paper yields a purely microscopic theory of optical pumping in laser physics. The model we consider is general enough to describe gauge invariant atom–reservoir interactions.
PubDate: 2015-06-01

• The Coefficient Problem and Multifractality of Whole-Plane SLE &amp;
LLE
• Abstract: Abstract Karl Löwner (later known as Charles Loewner) introduced his famous differential equation in 1923 to solve the Bieberbach conjecture for series expansion coefficients of univalent analytic functions at level n = 3. His method was revived in 1999 by Oded Schramm when he introduced the Stochastic Loewner Evolution (SLE), a conformally invariant process which made it possible to prove many predictions from conformal field theory for critical planar models in statistical mechanics. The aim of this paper is to revisit the Bieberbach conjecture in the framework of SLE processes and, more generally, Lévy processes. The study of their unbounded whole-plane versions leads to a discrete series of exact results for the expectations of coefficients and their variances, and, more generally, for the derivative moments of some prescribed order p. These results are generalized to the “oddified” or m-fold conformal maps of whole-plane SLEs or Lévy–Loewner Evolutions. We also study the (average) integral means multifractal spectra of these unbounded whole-plane SLE curves. We prove the existence of a phase transition at a moment order p = p *(κ) > 0, at which one goes from the bulk SLE κ average integral means spectrum, as predicted by the first author (Duplantier Phys. Rev. Lett. 84:1363–1367, 2000) and established by Beliaev and Smirnov (Commun Math Phys 290:577–595, 2009) and valid for p ≤ p *(κ), to a new integral means spectrum for p ≥ p *(κ), as conjectured in part by Loutsenko (J Phys A Math Gen 45(26):265001, 2012). The latter spectrum is, furthermore, shown to be intimately related, via the associated packing spectrum, to the radial SLE derivative exponents obtained by Lawler, Schramm and Werner (Acta Math 187(2):237–273, 2001), and to the local SLE tip multifractal exponents obtained from quantum gravity by the first author (Duplantier Proc. Sympos. Pure Math. 72(2):365–482, 2004). This is generalized to the integral means spectrum of the m-fold transform of the unbounded whole-plane SLE map. A succinct, preliminary, version of this study first appeared in Duplantier et al. (Coefficient estimates for whole-plane SLE processes, Hal-00609774, 2011).
PubDate: 2015-06-01

• Sharp Embedding of Sobolev Spaces Involving General Kernels and its
Application
• Abstract: Abstract The purpose of this paper is to extend the embedding theorem of Sobolev spaces involving general kernels and we provide a sharp critical exponent in the embedding. As an application, solutions for equations driven by a general integro-differential operator, with homogeneous Dirichlet boundary conditions, is established by using the Mountain Pass Theorem.
PubDate: 2015-06-01

• The Weyl Symbol of Schrödinger Semigroups
• Abstract: Abstract In this paper, we study the Weyl symbol of the Schrödinger semigroup e−tH , H = −Δ + V, t > 0, on $${L^2(\mathbb{R}^n)}$$ , with nonnegative potentials V in $${L^1_{\rm loc}}$$ . Some general estimates like the L ∞ norm concerning the symbol u are derived. In the case of large dimension, typically for nearest neighbor or mean field interaction potentials, we prove estimates with parameters independent of the dimension for the derivatives $${\partial_x^\alpha\partial_\xi^\beta u}$$ . In particular, this implies that the symbol of the Schrödinger semigroups belongs to the class of symbols introduced in Amour et al. (To appear in Proceedings of the AMS) in a high-dimensional setting. In addition, a commutator estimate concerning the semigroup is proved.
PubDate: 2015-06-01

• Overlapping Resonances in Open Quantum Systems
• Abstract: Abstract An N-level quantum system is coupled to a bosonic heat reservoir at positive temperature. We analyze the system–reservoir dynamics in the following regime: the strength λ of the system–reservoir coupling is fixed and small, but larger than the spacing σ of system energy levels. For vanishing σ there is a manifold of invariant system–reservoir states and for σ > 0 the only invariant state is the joint equilibrium. The manifold is invariant for σ = 0 but becomes quasi-invariant for σ > 0. Namely, on a first time-scale of the order 1/λ2, initial states approach the manifold. Then, they converge to the joint equilibrium state on a much larger time-scale of the order λ2/σ 2. We give a detailed expansion of the system–reservoir evolution showing the above scenario.
PubDate: 2015-06-01

• On the Probabilistic Nature of Quantum Mechanics and the Notion of Closed
Systems
• Abstract: Abstract The notion of “closed systems” in Quantum Mechanics is discussed. For this purpose, we study two models of a quantum mechanical system P spatially far separated from the “rest of the universe” Q. Under reasonable assumptions on the interaction between P and Q, we show that the system P behaves as a closed system if the initial state of P ∨ Q belongs to a large class of states, including ones exhibiting entanglement between P and Q. We use our results to illustrate the non-deterministic nature of quantum mechanics. Studying a specific example, we show that assigning an initial state and a unitary time evolution to a quantum system is generally not sufficient to predict the results of a measurement with certainty.
PubDate: 2015-05-30

• A Renormalization Group Method by Harmonic Extensions and the Classical
Dipole Gas
• Abstract: Abstract In this paper, we develop a new renormalization group method, which is based on conditional expectations and harmonic extensions, to study functional integrals of small perturbations of Gaussian fields. In this new method, one integrates Gaussian fields inside domains at all scales conditioning on the fields outside these domains, and by the variation principle solves local elliptic problems. It does not rely on an a priori decomposition of the Gaussian covariance. We apply this method to the model of classical dipole gas on the lattice, and show that the scaling limit of the generating function with smooth test functions is the generating function of the renormalized Gaussian free field.
PubDate: 2015-05-30

• On the K -Theoretic Classification of Topological Phases of Matter
• Abstract: Abstract We present a rigorous and fully consistent K-theoretic framework for studying gapped phases of free fermions. It utilizes and profits from powerful techniques in operator K-theory, which from the point of view of symmetries such as time reversal, charge conjugation, and magnetic translations, is more general and natural than the topological version. In our model-independent approach, the dynamics are only constrained by the physical symmetries, which can be completely encoded using a suitable C *-superalgebra. Contrary to existing literature, we do not use K-theory groups to classify phases in an absolute sense, but to classify topological obstructions between phases. The Periodic Table of Kitaev is exhibited as a special case within our framework, and we prove that the phenomena of periodicity and dimension shifts are robust against disorder and magnetic fields.
PubDate: 2015-05-28

• An Alexandrov–Fenchel-Type Inequality in Hyperbolic Space with an
Application to a Penrose Inequality
• Abstract: Abstract We prove a sharp Alexandrov–Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic n-space, n ≥ 3. The argument uses two new monotone quantities along the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter–Schwarzschild solution. This sharpens previous results by Dahl–Gicquaud–Sakovich and settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space–times with negative cosmological constant. We also explain how our methods can be easily adapted to derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension n ≥ 3. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chruściel and Simon on the validity of a Penrose-type inequality for exotic black holes.
PubDate: 2015-05-26

• Volume Comparison of Conformally Compact Manifolds with Scalar Curvature R
≥ − n ( n − 1)
• Abstract: Abstract In this paper, we use the normalized Ricci–DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature R ≥ −n (n − 1) and also the rigidity result when certain relative volume is zero.
PubDate: 2015-05-03

• Note on the Set of Bragg Peaks with High Intensity
• Abstract: Abstract We consider diffraction of Delone sets in Euclidean space. We show that the set of Bragg peaks with high intensity is always Meyer (if it is relatively dense). We use this to provide a new characterization for Meyer sets in terms of positive and positive definite measures. Our results are based on a careful study of positive definite measures, which may be of interest in its own right.
PubDate: 2015-05-03

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