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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  [2209 journals]   [SJR: 0.977]   [H-I: 26]
• Local Exclusion and Lieb–Thirring Inequalities for Intermediate and
Fractional Statistics
• Abstract: Abstract In one and two spatial dimensions there is a logical possibility for identical quantum particles different from bosons and fermions, obeying intermediate or fractional (anyon) statistics. We consider applications of a recent Lieb–Thirring inequality for anyons in two dimensions, and derive new Lieb–Thirring inequalities for intermediate statistics in one dimension with implications for models of Lieb–Liniger and Calogero–Sutherland type. These inequalities follow from a local form of the exclusion principle valid for such generalized exchange statistics.
PubDate: 2014-06-01

• Continuum Schrödinger Operators Associated With Aperiodic Subshifts
• Abstract: Abstract We study Schrödinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the standard theory that shows that the spectrum and the spectral type are almost surely constant and that identifies the almost sure absolutely continuous spectrum with the essential closure of the set of energies with vanishing Lyapunov exponent. Using results of Damanik–Lenz and Klassert–Lenz–Stollmann, we also show that the spectrum is a Cantor set of zero Lebesgue measure if the subshift satisfies the Boshernitzan condition and the potentials are aperiodic and irreducible. We then study the case of the Fibonacci subshift in detail and prove results for the local Hausdorff dimension of the spectrum at a given energy in terms of the value of the associated Fricke–Vogt invariant. These results are elucidated for some simple choices of the local potential pieces, such as piecewise constant ones and local point interactions. In the latter special case, our results explain the occurrence of so-called pseudo bands, which have been pointed out in the physics literature.
PubDate: 2014-06-01

• On Applications of Orlicz Spaces to Statistical Physics
• Abstract: Abstract We present a new rigorous approach based on Orlicz spaces for the description of the statistics of large regular statistical systems, both classical and quantum. The pair of Orlicz spaces we explicitly use are, respectively, built on the exponential function (for the description of regular observables) and on an entropic type function (for the corresponding states). They form a dual pair (both for classical and quantum systems). This pair has the advantage of being general enough to encompass regular observables, and specific enough for the latter Orlicz space to select states with a well-defined entropy function. Moreover for small quantum systems, this pair is shown to agree with the classical pairing of bounded linear operators on a Hilbert space, and the trace-class operators.
PubDate: 2014-06-01

• A New Asymptotic Perturbation Theory with Applications to Models of
Massless Quantum Fields
• Abstract: Let H 0 and H I be a self-adjoint and a symmetric operator on a complex Hilbert space, respectively, and suppose that H 0 is bounded below and the infimum E 0 of the spectrum of H 0 is a simple eigenvalue of H 0 which is not necessarily isolated. In this paper, we present a new asymptotic perturbation theory for an eigenvalue E(λ) of the operator ${H(\lambda)\,:=\,H_0 + \lambda H_{I}\,(\lambda \in \mathbb{R} \setminus \{0\})}$ satisfying lim λ → 0 E(λ) = E 0. The point of the theory is in that it covers also the case where E 0 is a non-isolated eigenvalue of H 0. Under a suitable set of assumptions, we derive an asymptotic expansion of E(λ) up to an arbitrary finite order of λ as λ → 0. We apply the abstract results to a model of massless quantum fields, called the generalized spin-boson model (Arai and Hirokawa in J Funct Anal 151:455–503, 1997) and show that the ground-state energy of the model has asymptotic expansions in the coupling constant λ as λ → 0.
PubDate: 2014-06-01

• Existence and Nonlinear Stability of Stationary States for the
Semi-Relativistic Schrödinger-Poisson System
• Abstract: Abstract We study the stationary states of the semi-relativistic Schrödinger-Poisson system in the repulsive (plasma physics) Coulomb case. In particular, we establish the existence and the nonlinear stability of a wide class of stationary states by means of the energy-Casimir method. We generalize the global well-posedness result of (Abou Salem et al. in Dyn Partial Differ Equ 9(2):121–132, 2012) for the semi-relativistic Schrödinger-Poisson system to spaces with higher regularity.
PubDate: 2014-06-01

• Multipolarons in a Constant Magnetic Field
• Abstract: Abstract The binding of a system of N polarons subject to a constant magnetic field of strength B is investigated within the Pekar–Tomasevich approximation. In this approximation, the energy of N polarons is described in terms of a non-quadratic functional with a quartic term that accounts for the electron–electron self-interaction mediated by phonons. The size of a coupling constant, denoted by α, in front of the quartic term is determined by the electronic properties of the crystal under consideration, but in any case it is constrained by 0 < α < 1. For all values of N and B, we find an interval α N,B <  α <  1 where the N polarons bind in a single cluster described by a minimizer of the Pekar–Tomasevich functional. This minimizer is exponentially localized in the N-particle configuration space ${\mathbb{R}^{3N}}$ .
PubDate: 2014-06-01

• Absence of Absolutely Continuous Spectrum for the Kirchhoff Laplacian on
• Abstract: Abstract In this paper, we prove that the existence of absolutely continuous spectrum of the Kirchhoff Laplacian on a radial metric tree graph together with a finite complexity of the geometry of the tree implies that the tree is in fact eventually periodic. This complements the results by Breuer and Frank in (Rev Math Phys 21(7):929–945, 2009) in the discrete case as well as for sparse trees in the metric case.
PubDate: 2014-06-01

• An Expansion Estimate for Dispersing Planar Billiards with Corner Points
• Abstract: Abstract It is known that the dynamics of planar billiards satisfies strong mixing properties (e.g., exponential decay of correlations), provided that some expansion condition on unstable curves is satisfied. This condition has been shown to always hold for smooth dispersing planar billiards, but it needed to be assumed separately in the case of dispersing planar billiards with corner points. We prove that this expansion condition holds for any dispersing planar billiard with corner points, no cusps and bounded horizon.
PubDate: 2014-06-01

• 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: 2014-05-04

• 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 ${E\in \mathbb{R}}$ ...
PubDate: 2014-05-04

• 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: 2014-05-01

• 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: 2014-05-01

• Self-Adjoint Extensions of Discrete Magnetic Schrödinger Operators
• Abstract: Abstract Using the concept of an intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schrödinger operator to be essentially self-adjoint. The present paper is an extension of some recent results proven in the context of graphs of bounded degree.
PubDate: 2014-05-01

• The 1/N Expansion of Multi-Orientable Random Tensor Models
• Abstract: Abstract Multi-orientable group field theory (GFT) was introduced in Tanasa (J Phys A 45:165401, 2012), as a quantum field theoretical simplification of GFT, which retains a larger class of tensor graphs than the colored one. In this paper we define the associated multi-orientable identically independent distributed multi-orientable tensor model and we derive its 1/N expansion. In order to obtain this result, a partial classification of general tensor graphs is performed and the combinatorial notion of jacket is extended to the m.o. graphs. We prove that the leading sector is given, as in the case of colored models, by the so-called melon graphs.
PubDate: 2014-05-01

• Renormalizability Conditions for Almost-Commutative Manifolds
• Abstract: Abstract We formulate conditions under which the asymptotically expanded spectral action on an almost-commutative manifold is renormalizable as a higher-derivative gauge theory. These conditions are of graph theoretical nature, involving the Krajewski diagrams that classify such manifolds. This generalizes our previous result on (super) renormalizability of the asymptotically expanded Yang–Mills spectral action to a more general class of particle-physics models that can be described geometrically in terms of a noncommutative space. In particular, it shows that the asymptotically expanded spectral action which at lowest order gives the Standard Model of elementary particles is renormalizable.
PubDate: 2014-05-01

• Modified Wave Operators Without Loss of Regularity for Some Long-Range
Hartree Equations: I
• Abstract: Abstract We reconsider the theory of scattering for some long-range Hartree equations with potential x −γ with 1/2 <  γ <  1. More precisely we study the local Cauchy problem with infinite initial time, which is the main step in the construction of the modified wave operators. We solve that problem in the whole subcritical range without loss of regularity between the asymptotic state and the solution, thereby recovering a result of Nakanishi. Our method starts from a different parametrization of the solutions, already used in our previous papers. This reduces the proofs to energy estimates and avoids delicate phase estimates.
PubDate: 2014-05-01

• On a Coordinate-Independent Description of String Worldsheet Theory
• Abstract: Abstract We rewrite the bosonic worldsheet theory in curved background in a language where it describes a single particle moving in an infinite-dimensional curved spacetime. This language is developed at a formal level without regularizing the infinite-dimensional traces. Then, we adopt DeWitt’s (Phys Rev 85:653, 1952) coordinate-independent formulation of quantum mechanics in the present context. This procedure enables us to define coordinate invariant quantum analogue of classical Virasoro generators, which we call DeWitt–Virasoro generators. This framework also enables us to calculate the invariant matrix elements of an arbitrary operator constructed out of the DeWitt–Virasoro generators between two arbitrary scalar states. Using these tools, we further calculate the DeWitt–Virasoro algebra in spin-zero representation. The result is given by the Witt algebra with additional anomalous terms that vanish for Ricci-flat backgrounds. Further analysis need to be performed to precisely relate this with the beta function computation of Friedan and others. Finally, we explain how this analysis improves the understanding of showing conformal invariance for certain pp-wave that has been recently discussed using hamiltonian framework.
PubDate: 2014-05-01

• The Mass Shell in the Semi-Relativistic Pauli–Fierz Model
• Abstract: Abstract We consider the semi-relativistic Pauli–Fierz model for a single free electron interacting with the quantized radiation field. Employing a variant of Pizzo’s iterative analytic perturbation theory we construct a sequence of ground state eigenprojections of infra-red cutoff, dressing transformed fiber Hamiltonians and prove its convergence, as the cutoff goes to zero. Its limit is the ground state eigenprojection of a certain renormalized fiber Hamiltonian. The ground state energy is an exactly twofold degenerate eigenvalue of the renormalized Hamiltonian, while it is not an eigenvalue of the original fiber Hamiltonian unless the total momentum is zero. These results hold true, for total momenta inside a ball about zero of arbitrary radius ${\mathfrak{p} > 0}$ , provided that the coupling constant is sufficiently small depending on ${\mathfrak{p}}$ and the ultra-violet cutoff. Along the way we prove twice continuous differentiability and strict convexity of the ground state energy as a function of the total momentum inside that ball.
PubDate: 2014-05-01

• Fixed Points of Compact Quantum Groups Actions on Cuntz Algebras
• Abstract: Abstract Given an action of a Compact Quantum Group (CQG) on a finite dimensional Hilbert space, we can construct an action on the associated Cuntz algebra. We study the fixed point algebra of this action, using Kirchberg classification results. Under certain conditions, we prove that the fixed point algebra is purely infinite and simple. We further identify it as a C *-algebra, compute its K-theory and prove a “stability property”: the fixed points only depend on the CQG via its fusion rules. We apply the theory to SU q (N) and illustrate by explicit computations for SU q (2) and SU q (3). This construction provides examples of free actions of CQG (or “principal noncommutative bundles”).
PubDate: 2014-05-01

• Next-to-Leading Order in the Large        class="a-plus-plus">N Expansion of the
Multi-Orientable Random Tensor Model
• Abstract: Abstract In this paper we analyze in detail the next-to-leading order (NLO) of the recently obtained large N expansion for the multi-orientable (MO) tensor model. From a combinatorial point of view, we find the class of Feynman tensor graphs contributing to this order in the expansion. Each such NLO graph is characterized by the property that it contains a certain non-orientable ribbon subgraph (a non-orientable jacket). Furthermore, we find the radius of convergence and the susceptibility exponent of the NLO series for this model. These results represent a first step towards the larger goal of defining an appropriate double-scaling limit for the MO tensor model.
PubDate: 2014-04-29

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