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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  [2276 journals]
• Deformations of Charged Axially Symmetric Initial Data and the
Mass–Angular Momentum–Charge Inequality
• Abstract: Abstract We show how to reduce the general formulation of the mass–angular momentum–charge inequality, for axisymmetric initial data of the Einstein–Maxwell equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. It is also 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, angular momentum, and charge. This extends previous work by the authors (Cha and Khuri, Ann Henri Poincaré, doi:10.1007/s00023-014-0332-6, arXiv:1401.3384, 2014), in which the role of charge was omitted. Lastly, we improve upon the hypotheses required for the mass–angular momentum–charge inequality in the maximal case.
PubDate: 2015-12-01

• Exact Results for a Toy Model Exhibiting Dynamic Criticality
• Abstract: Abstract We discuss a one-dimensional, periodic charge density wave model, and a toy model introduced in Kaspar and Mungan (EPL 103:46002, 2013) which is intended to approximate the former in the case of strong pinning. For both systems an external force may be applied, driving it in one of two (±) directions, and we describe an avalanche algorithm producing an ordered sequence of static configurations leading to the depinning thresholds. For the toy model these threshold configurations are explicit functions of the underlying quenched disorder, as is the threshold-to-threshold evolution via iteration of the algorithm. These explicit descriptions are used to study the law of the random polarization P for the toy model in two cases. Evolving from a macroscopically flat initial state to threshold, we give a scaling limit characterization determines the final value of P. Evolving from (−)-threshold to (+)-threshold, we use an identification with record sequences to study P as a function of the difference between the current force F and the threshold force F th. The results presented are rigorous and give strong evidence that the depinning transition in the toy model is a dynamic critical phenomenon.
PubDate: 2015-12-01

• Instability of Pre-Existing Resonances Under a Small Constant Electric
Field
• Abstract: Abstract Two simple model operators are considered which have pre-existing resonances. A potential corresponding to a small electric field, f, is then introduced and the resonances of the resulting operator are considered as f → 0. It is shown that these resonances are not continuous in this limit. It is conjectured that a similar behavior will appear in more complicated models of atoms and molecules. Numerical results are presented.
PubDate: 2015-12-01

• Conformal Parameterizations of Slices of Flat Kasner Spacetimes
• Abstract: Abstract The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method’s construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn our attention to non-CMC slices of the smaller family of flat Kasner spacetimes. In this restricted setting we obtain a full description of the construction of certain U n-1 symmetric slices, even in the far-from-CMC regime. Among the conformal data sets generating these slices we find that most data sets construct a single flat Kasner spacetime, but that there are also far-from-CMC data sets that construct one-parameter families of slices. Although these non-CMC families are analogues of well-known CMC one-parameter families, they differ in important ways. Most significantly, unlike the CMC case, the condition signaling the appearance of these non-CMC families is not naturally detected from the conformal data set itself. In light of this difficulty, we propose modifications of the conformal method that involve a conformally transforming mean curvature.
PubDate: 2015-12-01

• Rational Differential Systems, Loop Equations, and Application to the q th
Reductions of KP
• Abstract: Abstract To any solution of a linear system of differential equations, we associate a matrix kernel, correlators satisfying a set of loop equations, and in the presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight $${\hbar}$$ per derivative) of these quantities. When this expansion is of topological type (TT), the coefficients of expansions are computed by the topological recursion with initial data given by the semiclassical spectral curve of the linear system. This provides an efficient algorithm to compute them at least when the semiclassical spectral curve is of genus 0. TT is a non-trivial property, and it is an open problem to find a criterion which guarantees it is satisfied. We prove TT and illustrate our construction for the linear systems associated to the qth reductions of KP—which contain the (p, q) models as a specialization.
PubDate: 2015-12-01

• Differential Equations with Infinitely Many Derivatives and the Borel
Transform
• Abstract: Abstract Differential equations with infinitely many derivatives, sometimes also referred to as “nonlocal” differential equations, appear frequently in branches of modern physics such as string theory, gravitation and cosmology. We properly interpret and solve linear equations in this class with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous studies of nonlocal equations via Laplace and Fourier transforms, see for instance (Barnaby and Kamran, J High Energy Phys 02:40, 2008; Górka et al., Class Quantum Gravity 29:065017, 2012; Górka et al., Ann Henri Poincaré 14:947–966, 2013). We reconsider “generalized” initial value problems within the present approach and we disprove various conjectures found in modern physics literature. We illustrate various analytic phenomena that can occur with concrete examples, and we also treat efficient implementations of the theory.
PubDate: 2015-11-12

• Nonequilibrium Statistical Mechanics of Weakly Stochastically Perturbed
System of Oscillators
• Abstract: Abstract We consider a finite region of a d-dimensional lattice, $${d \in \mathbb{N}}$$ , of weakly coupled harmonic oscillators. The coupling is provided by a nearest-neighbour potential (harmonic or not) of size $${\varepsilon}$$ . Each oscillator weakly interacts by force of order $${\varepsilon}$$ with its own stochastic Langevin thermostat of arbitrary positive temperature. We investigate limiting as $${\varepsilon \rightarrow 0}$$ behaviour of solutions of the system and of the local energy of oscillators on long-time intervals of order $${\varepsilon^{-1}}$$ and in a stationary regime. We show that it is governed by an effective equation which is a dissipative SDE with nondegenerate diffusion. Next, we assume that the interaction potential is of size $${\varepsilon \lambda}$$ , where $${\lambda}$$ is another small parameter, independent from $${\varepsilon}$$ . Solutions corresponding to this scaling describe small low temperature oscillations. We prove that in a stationary regime, under the limit $${\varepsilon \rightarrow 0}$$ , the main order in $${\lambda}$$ of the averaged Hamiltonian energy flow is proportional to the gradient of temperature. We show that the coefficient of proportionality, which we call the conductivity, admits a representation through stationary space–time correlations of the energy flow. Most of the results and convergences we obtain are uniform with respect to the number of oscillators in the system.
PubDate: 2015-11-04

• Generalised Quantum Waveguides
• Abstract: Abstract We study general quantum waveguides and establish explicit effective Hamiltonians for the Laplacian on these spaces. A conventional quantum waveguide is an $${\epsilon}$$ -tubular neighbourhood of a curve in $${\mathbb{R}^3}$$ and the object of interest is the Dirichlet Laplacian on this tube in the asymptotic limit $${\epsilon\ll1}$$ . We generalise this by considering fibre bundles M over a complete d-dimensional submanifold $${B\subset\mathbb{R}^{d+k}}$$ with fibres diffeomorphic to $${F\subset\mathbb{R}^k}$$ , whose total space is embedded into an $${\epsilon}$$ -neighbourhood of B. From this point of view, B takes the role of the curve and F that of the disc-shaped cross section of a conventional quantum waveguide. Our approach allows, among other things, for waveguides whose cross sections F are deformed along B and also the study of the Laplacian on the boundaries of such waveguides. By applying recent results on the adiabatic limit of Schrödinger operators on fibre bundles we show, in particular, that for small energies the dynamics and the spectrum of the Laplacian on M are reflected by the adiabatic approximation associated with the ground state band of the normal Laplacian. We give explicit formulas for the accordingly effective operator on L 2(B) in various scenarios, thereby improving and extending many of the known results on quantum waveguides and quantum layers in $${\mathbb{R}^3}$$ .
PubDate: 2015-11-01

• Spectral Properties of Non-Unitary Band Matrices
• Abstract: Abstract We consider families of random non-unitary contraction operators defined as deformations of CMV matrices which appear naturally in the study of random quantum walks on trees or lattices. We establish several deterministic and almost sure results about the location and nature of the spectrum of such non-normal operators as a function of their parameters. We relate these results to the analysis of certain random quantum walks, the dynamics of which can be studied by means of iterates of such random non-unitary contraction operators.
PubDate: 2015-11-01

• Natural Extensions of Electroweak Geometry and Higgs Interactions
• Abstract: Abstract We explore the possibility that the Higgs boson of the standard model be actually a member of a larger family, by showing that a more elaborate internal structure naturally arises from geometrical arguments, in the context of a partly original handling of gauge fields which was put forward in previous papers. A possible mechanism yielding the usual Higgs potential is proposed. New types of point interactions, arising in particular from two-spinor index contractions, are shown to be allowed.
PubDate: 2015-11-01

• On Quantum Percolation in Finite Regular Graphs
• Abstract: Abstract The aim of this paper is twofold. First, we study eigenvalues and eigenvectors of the adjacency matrix of a bond percolation graph when the base graph is finite and well approximated locally by an infinite regular graph. We relate quantitatively the empirical measure of the eigenvalues and the delocalization of the eigenvectors to the spectrum of the adjacency operator of the percolation on the infinite graph. Secondly, we prove that percolation on an infinite regular tree with degree at least three preserves the existence of an absolutely continuous spectrum if the removal probability is small enough. These two results are notably relevant for bond percolation on a uniformly sampled regular graph or a Cayley graph with large girth.
PubDate: 2015-11-01

• The Translation Invariant Massive Nelson Model: III. Asymptotic
Completeness Below the Two-Boson Threshold
• Abstract: Abstract We show asymptotic completeness of two-body scattering for a class of translation invariant models describing a single quantum particle (the electron) linearly coupled to a massive scalar field (bosons). Our proof is based on a recently established Mourre estimate for these models. In contrast to previous approaches, it requires no number cutoff, no restriction on the particle–field coupling strength, and no restriction on the magnitude of total momentum. Energy, however, is restricted by the two-boson threshold, admitting only scattering of a dressed electron and a single asymptotic boson. The class of models we consider includes the UV-cutoff Nelson and polaron models. Although this paper is a part of a larger investigation, the presentation is self-contained.
PubDate: 2015-11-01

• Inverse Scattering at High Energies for Classical Relativistic Particles
in a Long-Range Electromagnetic Field
• Abstract: Abstract We define scattering data for the relativistic Newton equation in a static external electromagnetic field $${(-\nabla V, B)\in C^1(\mathbb{R}^n,\mathbb{R}^n)\times C^1(\mathbb{R}^n,A_n(\mathbb{R})), n\geq 2}$$ , that decays at infinity like $${r^{-\alpha-1}}$$ for some $${\alpha\in (0,1]}$$ , where $${A_n(\mathbb{R})}$$ is the space of $${n\times n}$$ antisymmetric matrices. We prove, in particular, that the short-range part of $${(\nabla V,B)}$$ can be reconstructed from the high-energy asymptotics of the scattering data provided that the long-range tail of $${(\nabla V,B)}$$ is known. We consider also inverse scattering in other asymptotic regimes. This work generalizes [Jollivet (Asympt Anal 55:103–123, 2007)] where a short-range electromagnetic field was considered.
PubDate: 2015-11-01

• Lieb–Robinson Bounds, Arveson Spectrum and Haag–Ruelle
Scattering Theory for Gapped Quantum Spin Systems
• Abstract: Abstract We consider translation invariant gapped quantum spin systems satisfying the Lieb–Robinson bound and containing single-particle states in a ground state representation. Following the Haag–Ruelle approach from relativistic quantum field theory, we construct states describing collisions of several particles, and define the corresponding S-matrix. We also obtain some general restrictions on the shape of the energy–momentum spectrum. For the purpose of our analysis, we adapt the concepts of almost local observables and energy–momentum transfer (or Arveson spectrum) from relativistic QFT to the lattice setting. The Lieb–Robinson bound, which is the crucial substitute of strict locality from relativistic QFT, underlies all our constructions. Our results hold, in particular, in the Ising model in strong transverse magnetic fields.
PubDate: 2015-10-23

• Eigenvalue Asymptotics for a Schrödinger Operator with Non-Constant
Magnetic Field Along One Direction
• Abstract: Abstract We consider the discrete spectrum of the two-dimensional Hamiltonian H = H 0 + V, where H 0 is a Schrödinger operator with a non-constant magnetic field B that depends only on one of the spatial variables, and V is an electric potential that decays at infinity. We study the accumulation rate of the eigenvalues of H in the gaps of its essential spectrum. First, under certain general conditions on B and V, we introduce effective Hamiltonians that govern the main asymptotic term of the eigenvalue counting function. Further, we use the effective Hamiltonians to find the asymptotic behavior of the eigenvalues in the case where the potential V is a power-like decaying function and in the case where it is a compactly supported function, showing a semiclassical behavior of the eigenvalues in the first case and a non-semiclassical behavior in the second one. We also provide a criterion for the finiteness of the number of eigenvalues in the gaps of the essential spectrum of H.
PubDate: 2015-10-23

• Re-Gauging Groupoid, Symmetries and Degeneracies for Graph Hamiltonians
and Applications to the Gyroid Wire Network
• Abstract: Abstract We study a class of graph Hamiltonians given by a type of quiver representation to which we can associate (non)-commutative geometries. By selecting gauging data, these geometries are realized by matrices through an explicit construction or a Kan extension. We describe the changes in gauge via the action of a re-gauging groupoid. It acts via matrices that give rise to a noncommutative 2-cocycle and hence to a groupoid extension (gerbe). We furthermore show that automorphisms of the underlying graph of the quiver can be lifted to extended symmetry groups of re-gaugings. In the commutative case, we deduce that the extended symmetries act via a projective representation. This yields isotypical decompositions and super-selection rules. We apply these results to the primitive cubic, diamond, gyroid and honeycomb wire networks using representation theory for projective groups and show that all the degeneracies in the spectra are consequences of these enhanced symmetries. This includes the Dirac points of the G(yroid) and the honeycomb systems.
PubDate: 2015-10-20

• On the Center of Mass of Asymptotically Hyperbolic Initial Data Sets
• Abstract: Abstract We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minkowskian spacetimes (isolated systems). In particular, we unite—an altered version of—the approach based on Hamiltonian charges with an approach based on CMC-foliations near infinity. The newly defined center of mass transforms appropriately under changes of the asymptotic coordinates and evolves in the direction of an appropriately defined linear momentum under the Einstein evolution equations.
PubDate: 2015-10-20

• Convergence of Clock Processes and Aging in Metropolis Dynamics of a
Truncated REM
• Abstract: Abstract We study the aging behavior of a truncated version of the Random Energy Model evolving under Metropolis dynamics. We prove that the natural time-time correlation function defined through the overlap function converges to an arcsine law distribution function, almost surely in the random environment and in the full range of time scales and temperatures for which such a result can be expected to hold. This establishes that the dynamics ages in the same way as Bouchaud’s REM-like trap model, thus extending the universality class of the latter model. The proof relies on a clock process convergence result of a new type where the number of summands is itself a clock process. This reflects the fact that the exploration process of Metropolis dynamics is itself an aging process, governed by its own clock. Both clock processes are shown to converge to stable subordinators below certain critical lines in their time-scale and temperature domains, almost surely in the random environment.
PubDate: 2015-10-20

• On the Focusing Effect of Gravitational Waves
• Abstract: Abstract We show that a suitable choice of the initial data from the past null infinity of Christodoulou’s work on the formation of black holes (The formation of black holes in general relativity. Monographs in Mathematics, European Mathematical Society, Zürich, 2009) will lead to the formation of a near Schwarzschild region up to the past null infinity.
PubDate: 2015-10-16

• Remainder Estimates for the Long Range Behavior of the van der Waals
Interaction Energy
• Abstract: Abstract The van der Waals–London’s law, for a collection of atoms at large separation, states that their interaction energy is pairwise attractive and decays proportionally to one over their distance to the sixth. The first rigorous result in this direction was obtained by Lieb and Thirring (Phys Rev A 34(1):40–46, 1986), by proving an upper bound which confirms this law. Recently the van der Waals–London’s law was proven under some assumptions by Anapolitanos and Sigal (arXiv:1205.4652v2). Following the strategy of Anapolitanos and Sigal (arXiv:1205.4652v2) and reworking the approach appropriately, we prove estimates on the remainder of the interaction energy. Furthermore, using an appropriate test function, we prove an upper bound for the interaction energy, which is sharp to leading order. For the upper bound, our assumptions are weaker, the remainder estimates stronger and the proof is simpler. The upper bound, for the cases it applies, improves considerably the upper bound of Lieb and Thirring. Their bound holds in a much more general setting, however. Here we consider only spinless Fermions.
PubDate: 2015-10-16

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