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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
• Localization for Random Block Operators Related to the XY Spin Chain
• Abstract: Abstract We study a class of random block operators which appear as effective one-particle Hamiltonians for the anisotropic XY quantum spin chain in an exterior magnetic field given by an array of i.i.d. random variables. For arbitrary non-trivial single-site distribution of the magnetic field, we prove dynamical localization of these operators at non-zero energy.
PubDate: 2014-04-13

• Global Wellposedness for a Certain Class of Large Initial Data for the 3D
Navier–Stokes Equations
• Abstract: Abstract In this article, we consider a special class of initial data to the 3D Navier–Stokes equations on the torus, in which there is a certain degree of orthogonality in the components of the initial data. We showed that, under such conditions, the Navier–Stokes equations are globally wellposed. We also showed that there exists large initial data, in the sense of the critical norm ${B^{-1}_{\infty,\infty}}$ that satisfies the conditions that we considered.
PubDate: 2014-04-01

• Properties of 1D Classical and Quantum Ising Models: Rigorous Results
• Abstract: Abstract In this paper, we consider one-dimensional classical and quantum spin-1/2 quasi-periodic Ising chains, with two-valued nearest neighbor interaction modulated by a Fibonacci substitution sequence on two letters. In the quantum case, we investigate the energy spectrum of the Ising Hamiltonian, in presence of constant transverse magnetic field. In the classical case, we investigate and prove analyticity of the free energy function when the magnetic field, together with interaction strength couplings, is modulated by the same Fibonacci substitution (thus proving absence of phase transitions of any order at finite temperature). We also investigate the distribution of Lee–Yang zeros of the partition function in the complex magnetic field regime, and prove its Cantor set structure (together with some additional qualitative properties), thus providing a rigorous justification for the observations in some previous works. In both, quantum and classical models, we concentrate on the ferromagnetic class.
PubDate: 2014-04-01

• Conformal Operators on Weighted Forms; Their Decomposition and Null Space
on Einstein Manifolds
• Abstract: Abstract There is a class of Laplacian like conformally invariant differential operators on differential forms ${L^\ell_k}$ which may be considered as the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as factored polynomials in second-order differential operators. In the case that the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the ${L^\ell_k}$ in terms of the null spaces of mutually commuting second-order factors.
PubDate: 2014-04-01

• Explicit Solution of the (Quantum) Elliptic Calogero–Sutherland
Model
• Abstract: Abstract The elliptic Calogero–Sutherland model is a quantum many body system of identical particles moving on a circle and interacting via two body potentials proportional to the Weierstrass ${\wp}$ -function. It also provides a natural many-variable generalization of the Lamé equation. Explicit formulas for the eigenfunctions and eigenvalues of this model as infinite series are obtained, to all orders and for arbitrary particle numbers and coupling parameters. These eigenfunctions are an elliptic deformation of the Jack polynomials. The absolute convergence of these series is proved in special cases, including the two-particle (=Lamé) case for non-integer coupling parameters and sufficiently small elliptic deformation.
PubDate: 2014-04-01

• Towards an Operator-Algebraic Construction of Integrable Global Gauge
Theories
• Abstract: Abstract The recent construction of integrable quantum field theories on two-dimensional Minkowski space by operator-algebraic methods is extended to models with a richer particle spectrum, including finitely many massive particle species transforming under a global gauge group. Starting from a two-particle S-matrix satisfying the usual requirements (unitarity, Yang–Baxter equation, Poincaré and gauge invariance, crossing symmetry, . . .), a pair of relatively wedge-local quantum fields is constructed which determines the field net of the model. Although the verification of the modular nuclearity condition as a criterion for the existence of local fields is not carried out in this paper, arguments are presented that suggest it holds in typical examples such as non-linear O(N)   σ-models. It is also shown that for all models complying with this condition, the presented construction solves the inverse scattering problem by recovering the S-matrix from the model via Haag–Ruelle scattering theory, and a proof of asymptotic completeness is given.
PubDate: 2014-04-01

• Minami’s Estimate: Beyond Rank One Perturbation and Monotonicity
• Abstract: Abstract In this note, we prove Minami’s estimate for a class of discrete alloy-type models with a sign-changing single-site potential of finite support. We apply Minami’s estimate to prove Poisson statistics for the energy level spacing. Our result is valid for random potentials which are in a certain sense sufficiently close to the standard Anderson potential (rank one perturbations coupled with i.i.d. random variables).
PubDate: 2014-04-01

• Eigenvalue Estimates for Non-Selfadjoint Dirac Operators on the Real Line
• Abstract: Abstract We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L 1-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.
PubDate: 2014-04-01

• Semi-Classical Measures on Quantum Graphs and the Gauß Map of the
Determinant Manifold
• Abstract: Abstract In this paper, we describe the weak limits of the measures associated to the eigenfunctions of the Laplacian on a Quantum graph for a generic metric in terms of the Gauss map of the determinant manifold. We describe also all the limits with minimal support (the “scars”).
PubDate: 2014-03-28

• The Real Spectrum of the Imaginary Cubic Oscillator: An Expository Proof
• Abstract: Abstract We give a partially alternate proof of reality of the spectrum of the imaginary cubic oscillator in quantum mechanics.
PubDate: 2014-03-27

• The Scaling and Mass Expansion
• Abstract: Abstract The scaling and mass expansion (shortly ‘sm-expansion’) is a new axiom for causal perturbation theory, which is a stronger version of a frequently used renormalization condition in terms of Steinmann’s scaling degree (Brunetti et al. in Commun Math Phys 208:623–661, 2000, Epstein et al. in Ann Inst Henri Poincaré 19A:211–295, 1973). If one quantizes the underlying free theory by using a Hadamard function (which is smooth in m ≥  0), one can reduce renormalization of a massive model to the extension of a minimal set of mass-independent, almost homogeneously scaling distributions by a Taylor expansion in the mass m. The sm-expansion is a generalization of this Taylor expansion, which yields this crucial simplification of the renormalization of massive models also for the case that one quantizes with the Wightman two-point function, which contains a log(−(m 2(x 2 − ix 0 0))-term. We construct the general solution of the new system of axioms (i.e. the usual axioms of causal perturbation theory completed by the sm-expansion), and illustrate the method for a divergent diagram which contains a divergent subdiagram.
PubDate: 2014-03-25

• 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        class="a-plus-plus">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        id-i-eq1">        format-t-e-x">${\mathcal{N} = 1}$ SCFTs
• 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

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