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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
• Central Limit Theorems for Open Quantum Random Walks and Quantum Measurement Records
• Abstract: Abstract Open Quantum Random Walks, as developed in Attal et al. (J. Stat. Phys. 147(4):832–852, 2012), are a quantum generalization of Markov chains on finite graphs or on lattices. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in quantum information theory (such as the well-known Hadamard random walk). Typically, in the case of open quantum random walks on lattices, their distribution seems to always converge to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous open quantum random walks on ${\mathbb{Z}^{\rm d},}$ we prove such a central limit theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on ${\mathbb{Z}^{\rm d}}$ times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a central limit theorem with explicit drift and explicit covariance matrix. The interesting point which appears with our construction and result is that it applies actually to a wider setup: it provides a central limit theorem for the sequence of recordings of the quantum trajectories associated wih any completely positive map. This is what we show and develop as an application of our result. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block diagonal in a common basis.
PubDate: 2014-03-07

• Topology, Rigid Cosymmetries and Linearization Instability in Higher Gauge Theories
• Abstract: Abstract We consider a class of non-linear PDE systems, whose equations possess Noether identities (the equations are redundant), including non-variational systems (not coming from Lagrangian field theories), where Noether identities and infinitesimal gauge transformations need not be in bijection. We also include theories with higher stage Noether identities, known as higher gauge theories (if they are variational). Some of these systems are known to exhibit linearization instabilities: there exist exact background solutions about which a linearized solution is extendable to a family of exact solutions only if some non-linear obstruction functionals vanish. We give a general, geometric classification of a class of these linearization obstructions, which includes as special cases all known ones for relativistic field theories (vacuum Einstein, Yang–Mills, classical N = 1 supergravity, etc.). Our classification shows that obstructions arise due to the simultaneous presence of rigid cosymmetries (generalized Killing condition) and non-trivial de Rham cohomology classes (spacetime topology). The classification relies on a careful analysis of the cohomologies of the on-shell Noether complex (consistent deformations), adjoint Noether complex (rigid cosymmetries) and variational bicomplex (conserved currents). An intermediate result also gives a criterion for identifying non-linearities that do not lead to linearization instabilities.
PubDate: 2014-03-04

• Inverse-Closed Algebras of Integral Operators on Locally Compact Groups
• Abstract: Abstract We construct some inverse-closed algebras of bounded integral operators with operator-valued kernels, acting in spaces of vector-valued functions on locally compact groups. To this end we make use of covariance algebras associated to C*-dynamical systems defined by the C*-algebras of right uniformly continuous functions with respect to the left regular representation.
PubDate: 2014-03-02

• On the Partial Differential Equations of Electrostatic MEMS Devices with Effects of Casimir Force
• Abstract: Abstract We analyze pull-in instability of electrostatically actuated microelectromechanical systems, and we find that as the device size is reduced, the effect of the Casimir force becomes more important. In the miniaturization process there is a minimum size for the device below which the system spontaneously collapses with zero applied voltage. According to the mathematical analysis, we obtain a set U in the plane, such that elements of U correspond to minimal stable solutions of a two-parameter mathematical model. For points on the boundary ${\Upsilon}$ of U, there exists weak solutions to this model, which are called extremal solutions. More refined properties of stable solutions—such as regularity, stability, uniqueness—are also established.
PubDate: 2014-03-02

• Spectral Theory of Semibounded Schrödinger Operators with δ′-Interactions
• Abstract: Abstract We study spectral properties of Hamiltonians H X,β,q with δ′-point interactions on a discrete set ${X = \{x_k\}_{k=1}^\infty \subset (0, +\infty)}$ . Using the form approach, we establish analogs of some classical results on operators H q =  −d2/dx 2 + q with locally integrable potentials ${q \in L^1_{\rm loc}[0, +\infty)}$ . In particular, we establish the analogues of the Glazman–Povzner–Wienholtz theorem, the Molchanov discreteness criterion, and the Birman theorem on stability of an essential spectrum. It turns out that in contrast to the case of Hamiltonians with δ-interactions, spectral properties of operators H X,β,q are closely connected with those of ${{\rm H}_{X,q}^N = \oplus_{k}{\rm H}_{q,k}^N}$ , where ${{\rm H}_{q,k}^N}$ is the Neumann realization of −d2/dx 2 + q in L 2(x k-1,x k ).
PubDate: 2014-03-01

• Decorrelation Estimates for a 1D Tight Binding Model in the Localized Regime
• Abstract: Abstract In this article, we prove decorrelation estimates for the eigenvalues of a 1D discrete tight-binding model near two distinct energies in the localized regime. Consequently, for any integer n ≥ 2, the asymptotic independence for local level statistics near n distinct energies is obtained.
PubDate: 2014-03-01

• On the Spectrum of 1D Quantum Ising Quasicrystal
• Abstract: Abstract We consider one-dimensional quantum Ising spin-1/2 chains with two-valued nearest neighbor couplings arranged in a quasi-periodic sequence, with uniform, transverse magnetic field. By employing the Jordan–Wigner transformation of the spin operators to spinless fermions, the energy spectrum can be computed exactly on a finite lattice. By employing the transfer matrix technique and investigating the dynamics of the corresponding trace map, we show that in the thermodynamic limit the energy spectrum is a Cantor set of zero Lebesgue measure. Moreover, we show that local Hausdorff dimension is continuous and non-constant over the spectrum. This forms a rigorous counterpart of numerous numerical studies.
PubDate: 2014-03-01

• Derivation of the Cubic NLS and Gross–Pitaevskii Hierarchy from Manybody Dynamics in d = 3 Based on Spacetime Norms
• Abstract: Abstract We derive the defocusing cubic Gross–Pitaevskii (GP) hierarchy in dimension d = 3, from an N-body Schrödinger equation describing a gas of interacting bosons in the GP scaling, in the limit N → ∞. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies (Chen and Pavlović in Discr Contin Dyn Syst 27(2):715–739, 2010; http://arxiv.org/abs/0906.2984; Proc Am Math Soc 141:279–293, 2013), which are inspired by the solution spaces based on space-time norms introduced by Klainerman and Machedon (Comm Math Phys 279(1):169–185, 2008). We note that in d = 3, this has been a well-known open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schrödinger equation (NLS) in d = 3.
PubDate: 2014-03-01

• Wilson Loops in 5d ${\mathcal{N} = 1}$ SCFTs and AdS/CFT
• Abstract: Abstract We consider ${\frac{1}{2}}$ -BPS circular Wilson loops in a class of 5d superconformal field theories on S 5. The large N limit of the vacuum expectation values of Wilson loops are computed both by localization in the field theory and by evaluating the fundamental string and D4-brane actions in the dual massive IIA supergravity background. We find agreement in the leading large N limit for a rather general class of representations, including fundamental, anti-symmetric and symmetric representations. For single-node theories the match is straightforward, while for quiver theories, the Wilson loop can be in different representations for each node. We highlight the two special cases when the Wilson loop is in either in all symmetric or all anti-symmetric representations. In the anti-symmetric case, we find that the vacuum expectation value factorizes into distinct contributions from each quiver node. In the dual supergravity description, this corresponds to probe D4-branes wrapping internal S 3 cycles. The story is more complicated in the symmetric case and the vacuum expectation value does not exhibit factorization.
PubDate: 2014-03-01

• Null Structure in a System of Quadratic Derivative Nonlinear Schrödinger Equations
• Abstract: Abstract We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schrödinger equations in two space dimensions with the masses satisfying the resonance relation. We present a structural condition on the nonlinearity under which small data global existence holds. It is also shown that the solution is asymptotically free. Our proof is based on the commuting vector field method combined with smoothing effects.
PubDate: 2014-02-04

• Random-Weighted Sobolev Inequalities on ${\mathbb{R}^d }$ and Application to Hermite Functions
• Abstract: Abstract We extend a randomization method, introduced by Shiffman–Zelditch and developed by Burq–Lebeau on compact manifolds for the Laplace operator, to the case of ${\mathbb{R}^d}$ with the harmonic oscillator. We construct measures, thanks to probability laws which satisfies the concentration of measure property, on the support of which we prove optimal-weighted Sobolev estimates on ${\mathbb{R}^d}$ . This construction relies on accurate estimates on the spectral function in a non-compact configuration space. As an application, we show that there exists a basis of Hermite functions with good decay properties in ${L^{\infty}(\mathbb{R}^{d})}$ , when d ≥ 2.
PubDate: 2014-02-04

• Remarks on Local Symmetry Invariance in Perturbative Algebraic Quantum Field Theory
• Abstract: Abstract We investigate various aspects of invariance under local symmetries in the framework of perturbative algebraic quantum field theory (pAQFT). Our main result is the proof that the quantum Batalin–Vilkovisky operator, on-shell, can be written as the commutator with the interacting BRST charge. Up to now, this was proven only for a certain class of fields in quantum electrodynamics and in Yang–Mills theory. Our result is more general and it holds in a wide class of theories with local symmetries, including general relativity and the bosonic string. We also comment on other issues related to local gauge invariance and, using the language of homological algebra, we compare different approaches to quantization of gauge theories in the pAQFT framework.
PubDate: 2014-02-02

• Infraparticle Problem, Asymptotic Fields and Haag–Ruelle Theory
• Abstract: Abstract In this article, we want to argue that an appropriate generalization of the Wigner concepts may lead to an asymptotic particle with well-defined mass, although no mass hyperboloid in the energy–momentum spectrum exists.
PubDate: 2014-02-01

• Non-Equilibrium States of a Photon Cavity Pumped by an Atomic Beam
• Abstract: Abstract We consider a beam of two-level randomly excited atoms that pass one-by-one through a one-mode cavity. We show that in the case of an ideal cavity, i.e. no leaking of photons from the cavity, the pumping by the beam leads to an unlimited increase in the photon number in the cavity. We derive an expression for the mean photon number for all times. Taking into account leaking of the cavity, we prove that the mean photon number in the cavity stabilizes in time. The limiting state of the cavity in this case exists and it is independent of the initial state. We calculate the characteristic functional of this non-quasi-free non-equilibrium state. We also calculate the total energy variation in both the ideal and the open cavities as well as the entropy production in the ideal cavity.
PubDate: 2014-02-01

• Symmetries of Quantum Lax Equations for the Painlevé Equations
• Abstract: Abstract Based on the fact that the Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries, a canonical quantization of the Painlevé equations preserving such symmetries has been studied recently. On the other hand, since the Painlevé equations can also be described as isomonodromic deformations of certain second-order linear differential equations, a quantization of such Lax formalism is also a natural problem. In this paper, we introduce a canonical quantization of Lax equations for the Painlevé equations and study their symmetries. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.
PubDate: 2014-02-01

• Isoperimetric Inequalities for a Wedge-Like Membrane
• Abstract: Abstract For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for the fundamental eigenvalue which in some cases improves the classical isoperimetric inequality of Faber–Krahn. In this work, we introduce “relative torsional rigidity” for this type of membrane and prove new isoperimetric inequalities in the spirit of Saint-Venant, Pólya–Szegő, Payne, Payne–Rayner, Chiti, and Talenti, which link the eigenvalue problem with the boundary value problem in a fundamental way.
PubDate: 2014-02-01

• Complete Asymptotic Expansion of the Integrated Density of States of Multidimensional Almost-Periodic Pseudo-Differential Operators
• Abstract: Abstract We obtain a complete asymptotic expansion of the integrated density of states of operators of the form ${H = (-\Delta)^w+ B}$ in ${\mathbb{R}^d}$ . Here w >  0 and B belong to a wide class of almost-periodic self-adjoint pseudo-differential operators of order less than 2w. In particular, we obtain such an expansion for magnetic Schrödinger operators with either smooth periodic or generic almost-periodic coefficients.
PubDate: 2014-02-01

• Quantum Ergodicity for a Point Scatterer on the Three-Dimensional Torus
• Abstract: Abstract Consider a point scatterer (the Laplacian perturbed by a delta-potential) on the standard three-dimensional flat torus. Together with the eigenfunctions of the Laplacian which vanish at the point, this operator has a set of new, perturbed eigenfunctions. In a recent paper, the author was able to show that all of the perturbed eigenfunctions are uniformly distributed in configuration space. In this paper we prove that almost all of these eigenfunctions are uniformly distributed in phase space, i.e. we prove quantum ergodicity for the subspace of the perturbed eigenfunctions. An analogue result for a point scatterer on the two-dimensional torus was recently proved by Kurlberg and Ueberschär.
PubDate: 2014-02-01

• Non-Existence of Toroidal Cohomogeneity-1 Near-Horizon Geometries
• Abstract: Abstract We prove that D ≥  5 dimensional stationary, non-static near-horizon geometries with (D−3) commuting rotational symmetries subject to the vacuum Einstein equations including a cosmological constant cannot have toroidal horizon topology. In D =  4 dimensions, the same result is obtained under the assumption of a non-negative cosmological constant.
PubDate: 2014-02-01

• Erratum to: Mather Measures Associated with a Class of Bloch Wave Functions
• PubDate: 2013-12-17

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