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 Annales Henri Poincaré    [4 followers]  Follow        Hybrid journal (It can contain Open Access articles)      ISSN (Print) 1424-0637 - ISSN (Online) 1424-0661      Published by Springer-Verlag  [2208 journals]   [SJR: 0.977]   [H-I: 26]
• A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian
• Abstract: Abstract We consider the Landau Hamiltonian perturbed by a long-range electric potential V. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we estimate the rate of the shrinking of these clusters to the Landau levels as the number of the cluster tends to infinity. Further, we assume that there exists an appropriate $${\mathbb{V}}$$ , homogeneous of order −ρ with $${\rho \in (0, 1)}$$ , such that $${V(x) = \mathbb{V} (x) + O( x ^{-\rho - \varepsilon})}$$ , ɛ >  0, as x → ∞, and investigate the asymptotic distribution of the eigenvalues within the qth cluster as q → ∞. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the mean-value transform of $${\mathbb{V}}$$ .
PubDate: 2014-08-01

• Exponential Decay of Equal-Time Four-Point Correlation Functions in the
Hubbard Model on the Copper-Oxide Lattice
• Abstract: Abstract For the Hubbard model on the two-dimensional copper-oxide lattice, equal-time four-point correlation functions at positive temperature are proved to decay exponentially in the thermodynamic limit if the magnitude of the on-site interactions is smaller than some power of temperature. This result especially implies that the equal-time correlation functions for singlet Cooper pairs of various symmetries decay exponentially in the distance between the Cooper pairs in high temperatures or in low-temperature weak-coupling regimes. The proof is based on a multi-scale integration over the Matsubara frequency.
PubDate: 2014-08-01

• Resonant Delocalization on the Bethe Strip
• Abstract: Abstract Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schrödinger operators on the Bethe lattice through rare resonances (resonant delocalization). We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble. These operators can be viewed as random operators on the Bethe strip, a graph (lattice) with loops.
PubDate: 2014-08-01

• Hypergeometric Type Functions and Their Symmetries
• Abstract: Abstract The paper is devoted to a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F 1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, recurrence relations of their solutions, their integral representations and discrete symmetries are discussed.
PubDate: 2014-08-01

• 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: 2014-07-22

• 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: 2014-07-18

• 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: 2014-07-16

• Zero Modes of Quantum Graph Laplacians and an Index Theorem
• Abstract: Abstract We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper, the number of zero modes is expressed in terms of the trace of a unitary matrix $${\mathfrak{S}}$$ that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part, a Dirac operator is defined whose square is related to the Laplacian. To accommodate Laplacians with negative eigenvalues, it is necessary to define the Dirac operator on a suitable Kreĭn space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kreĭn-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator.
PubDate: 2014-07-11

• 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: 2014-07-10

• 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: 2014-07-09

• 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: 2014-07-09

• 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: 2014-07-08

• 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: 2014-07-05

• Initial Data Sets with Ends of Cylindrical Type: I. The Lichnerowicz
Equation
• Abstract: Abstract We construct large classes of vacuum general relativistic initial data sets, possibly with a cosmological constant $${\Lambda \in \mathbb{R}}$$ , containing ends of cylindrical type.
PubDate: 2014-07-02

• Gaussian Free Fields and KPZ Relation in        inline-equation id-i-eq1">        equation-source format-t-e-x">$${\mathbb{R}^4}$$

• Abstract: Abstract This work aims to extend part of the two-dimensional results of Duplantier and Sheffield on Liouville quantum gravity (Invent Math 185(2):333–393, 2011) to four dimensions, and indicate possible extensions to other even-dimensional spaces $${\mathbb{R}^{2n}}$$ as well as Riemannian manifolds. Let Θ be the Gaussian free field on $${\mathbb{R}^{4}}$$ with the underlying Hilbert space $${H^{2}\left(\mathbb{R}^{4}\right)}$$ and the inner product $${\left(\left(I-\Delta\right)^{2}\cdot,\cdot\right)_{L^{2}}}$$ , and θ a generic element from Θ. We consider a sequence of random Borel measures on $${\mathbb{R}^{4}}$$ , denoted by $${\left\{m_{\epsilon_{n}}^{\theta}\left({\rm d}x\right):n\geq1\right\}}$$ , each of which is absolutely continuous with respect to the Lebesgue measure dx, and the density function is given by the exponential of a centered Gaussian family parametrized by $${x \in \mathbb{R}^{4}}$$ . We show that with probability 1, $${m_{\epsilon_{n}}^{\theta}\left({\rm d}x\right)}$$ weakly converges as $${\epsilon_{n} \downarrow 0}$$ , and the limit measure can be “formally” written as “ $${m^{\theta}\left({\rm d}x\right) = e^{2\gamma\theta\left(x\right)}{\rm d}x}$$ ”. In this setting, we also prove a KPZ relation, which is the quadratic relation between the scaling exponent of a bounded Borel set on $${\mathbb{R}^{4}}$$ under the Lebesgue measure and its counterpart under the random measure $${m^{\theta}\left({\rm d}x\right)}$$ . Our approach is similar to the one used in Duplantier and Sheffield (Invent Math 185(2):333–393, 2011) with adaptations to $${\mathbb{R}^{4}}$$ .
PubDate: 2014-07-01

• On Second Variation of Wang–Yau Quasi-Local Energy
• Abstract: Abstract We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang–Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly round surfaces including large constant mean curvature spheres in asymptotically flat 3-manifolds with positive mass; it is also positive definite on small geodesics spheres, whose centers do not have vanishing curvature, in Riemannian 3-manifolds of nonnegative scalar curvature. We also give examples of functions H, which can be made arbitrarily close to 2, on the standard 2-sphere $${(\mathbb{S}^2, \sigma_0)}$$ such that the triple $${(\mathbb{S}^2, \sigma_0, H)}$$ has positive Brown–York mass while the associated functional is negative somewhere.
PubDate: 2014-07-01

• The Chaplygin Gas Equation of State for the Quantized Free Scalar Field on
Cosmological Spacetimes
• Abstract: Abstract In this paper, we try to answer the question whether the quantized free scalar field on a spatially flat Friedmann–Robertson–Walker (FRW) spacetime is a matter model that can induce a Chaplygin gas equation of state. For this purpose, we first describe how one can obtain every possible homogeneous and isotropic Hadamard (HIH) state once any such state is given. We also identify a condition on the scale factor sufficient to entail the existence of a simple HIH state—this state is constructed explicitly and can thence be used as a starting point for constructing all HIH states. Furthermore, we employ these results to show that on an FRW spacetime with non-positive constant scalar curvature, there is, with one exception, no Chaplygin gas equation of state compatible with any HIH state. Finally, we argue that the semi-classical Einstein equation and the Chaplygin gas equation of state can presumably not be consistently solved for the quantized free scalar field.
PubDate: 2014-07-01

• The Reduced Hartree–Fock Model for Short-Range Quantum Crystals with
Nonlocal Defects
• Abstract: Abstract In this article, we consider quantum crystals with defects in the reduced Hartree–Fock framework. The nuclei are supposed to be classical particles arranged around a reference periodic configuration. The perturbation is assumed to be small in amplitude, but need not be localized in a specific region of space or have any spatial invariance. Assuming Yukawa interactions, we prove the existence of an electronic ground state, solution of the self-consistent field equation. Next, by studying precisely the decay properties of this solution for local defects, we are able to expand the density of states of the nonlinear Hamiltonian of a system with a random perturbation of Anderson–Bernoulli type, in the limit of low concentration of defects. One important step in the proof of our results is the analysis of the dielectric response of the crystal to an effective charge perturbation.
PubDate: 2014-07-01

• A Class of Conformal Curves in the Reissner–Nordström Spacetime
• Abstract: Abstract A class of curves with special conformal properties (conformal curves) is studied on the Reissner–Nordström spacetime. It is shown that initial data for the conformal curves can be prescribed so that the resulting congruence of curves extends smoothly to future and past null infinity. The formation of conjugate points on these congruences is examined. The results of this analysis are expected to be of relevance for the discussion of the Reissner–Nordström spacetime as a solution to the conformal field equations and for the global numerical evaluation of static black hole spacetimes.
PubDate: 2014-07-01

• Hawking Effect for a Toy Model of Interacting Fermions
• Abstract: Abstract We consider a toy model of interacting Dirac fermions in a 1 + 1 dimensional space time describing the exterior of a star collapsing to a black hole. In this situation, we give a rigorous proof of the Hawking effect, namely that under the associated quantum evolution, an initial vacuum state will converge when t → + ∞ to a thermal state at Hawking temperature. We establish this result both for observables falling into the blackhole along null characteristics and for static observables. We also consider the case of an interaction localized near the star boundary, obtaining similar results. We hence extend to an interacting model previous results of Bachelot and Melnyk, obtained for free Dirac fields.
PubDate: 2014-06-28

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