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Annales Henri Poincaré

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ISSN (Print) 1424-0637 - ISSN (Online) 1424-0661
Published by Springer-Verlag

[2216 journals]

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: 2013-05-03
The Translation Invariant Massive Nelson Model: II. The Continuous Spectrum below the Two-Boson Threshold
- Abstract: Abstract In this paper we continue the study of the energy-momentum spectrum of a class of translation invariant, linearly coupled, and massive Hamiltonians from non-relativistic quantum field theory. The class contains the Hamiltonians of E. Nelson (J Math Phys 5:1190–1197, 1964) and H. Fröhlich (Adv Phys 3:325–362, 1954). In Møller (Ann Henri Poincaré 6:1091–1135, 2005; Rev Math Phys 18:485–517, 2006) one of us previously investigated the structure of the ground state mass shell and the bottom of the continuous energy-momentum spectrum. Here we study the continuous energy-momentum spectrum itself up to the two-boson threshold, the threshold for energetic support of two-boson scattering states. We prove that non-threshold embedded mass shells have finite multiplicity and can accumulate only at thresholds. We furthermore establish the non-existence of singular continuous energy-momentum spectrum. Our results hold true for all values of the particle-field coupling strength but only below the two-boson threshold. The proof revolves around the construction of a certain relative velocity vector field used to construct a conjugate operator in the sense of Mourre.
  PubDate: 2013-05-01
Remark on Itô’s Diffusion in Multidimensional Scattering with Sign-Indefinite Potentials
- Abstract: Abstract This paper extends some results of Denisov and Kupin (Int Math Res Not, doi:10.1093/imrn/rnr131, 2011) to the case of sign–indefinite potentials by applying methods developed in Denisov (J Funct Anal 254:2186–2226, 2008). This enables us to prove the presence of a.c. spectrum for the generic coupling constant.
  PubDate: 2013-05-01
Dynamical Locality of the Nonminimally Coupled Scalar Field and Enlarged Algebra of Wick Polynomials
- Abstract: Abstract We discuss dynamical locality in two locally covariant quantum field theories: the nonminimally coupled scalar field and the enlarged algebra of Wick polynomials. We calculate the relative Cauchy evolution of the enlarged algebra, before demonstrating that dynamical locality holds in the nonminimally coupled scalar field theory. We also establish dynamical locality in the enlarged algebra for the minimally coupled massive case and the conformally coupled massive case.
  PubDate: 2013-05-01
Existence of Wave Operators with Time-Dependent Modifiers for the Schrödinger Equations with Long-Range Potentials on Scattering Manifolds
- Abstract: Abstract We construct time-dependent wave operators for Schrödinger equations with long-range potentials on a manifold M with asymptotically conic structure. We use the two space scattering theory formalism, and a reference operator on a space of the form ${\mathbb{R} \times \partial M}$ , where ${\partial M}$ is the boundary of M at infinity. We construct exact solutions to the Hamilton–Jacobi equation on the reference system ${\mathbb{R} \times \partial M}$ and prove the existence of the modified wave operators.
  PubDate: 2013-05-01
Repeated Quantum Non-Demolition Measurements: Convergence and Continuous Time Limit
- Abstract: Abstract We analyze general enough models of repeated indirect measurements in which a quantum system interacts repeatedly with randomly chosen probes on which von Neumann direct measurements are performed. We prove, under suitable hypotheses, that the system state probability distribution converges after a large number of repeated indirect measurements, in a way compatible with quantum wave function collapse. We extend this result to mixed states and we prove similar results for the system density matrix. We show that the convergence is exponential with a rate given by some relevant mean relative entropies. We also prove that, under appropriate rescaling of the system and probe interactions, the state probability distribution and the system density matrix are solutions of stochastic differential equations modeling continuous-time quantum measurements. We analyze the large time behavior of these continuous time processes and prove convergence.
  PubDate: 2013-05-01
On a General Class of Nonlocal Equations
- Abstract: Abstract Motivated by recent developments in cosmology and string theory, we introduce a functional calculus appropriate for the study of non-linear nonlocal equations of the form f(Δ)u = U(x, u(x)) on Euclidean space. We prove that under some technical assumptions, these equations admit smooth solutions. We also consider nonlocal equations on compact Riemannian manifolds, and we prove the existence of smooth solutions. Moreover, in the Euclidean case we present conditions on f which guarantee that the solutions we find are, in fact, real-analytic.
  PubDate: 2013-05-01
Parametric Cutoffs for Interacting Fermi Liquids
- Abstract: Abstract This paper is a sequel to Disertori et al. (Annales Henri Poincaré 2, 733–806, 2001). We introduce a new multiscale decomposition of the Fermi propagator based on its parametric representation. We prove that the corresponding sliced propagator obeys the same direct space bounds than the decomposition used in Disertori et al. (Annales Henri Poincaré 2, 733–806, 2001). Therefore the non perturbative bounds on completely convergent contributions of Disertori et al. (Annales Henri Poincaré 2, 733–806, 2001) still hold. In addition the new slicing better preserves momenta, hence should become an important new technical tool for the rigorous analysis of condensed matter systems. In particular it should allow to complete the proof that a three dimensional interacting system of Fermions with spherical Fermi surface is a Fermi liquid in the sense of Salmhofer’s criterion.
  PubDate: 2013-05-01
Mass Operator and Dynamical Implementation of Mass Superselection Rule
- Abstract: Abstract We start reviewing Giulini’s dynamical approach to Bargmann superselection rule proposing some improvements. First of all we discuss some general features of the central extensions of the Galilean group used in Giulini’s programme, in particular focussing on the interplay of classical and quantum picture, without making any particular choice for the multipliers. Preserving other features of Giulini’s approach, we modify the mass operator of a Galilei invariant quantum system to obtain a mass spectrum that is (i) positive and (ii) discrete, so giving rise to a standard (non-continuous) superselection rule. The model results to be invariant under time reversal but a further degree of freedom appears that can be interpreted as describing an internal conserved charge of the system (however, adopting a POVM approach, the unobservable degrees of freedom can be pictured as a generalized observable automatically gaining a positive mass operator without assuming the existence of such a charge). The effectiveness of Bargmann rule is shown to be equivalent to an averaging procedure over the unobservable degrees of freedom of the central extension of Galileian group. Moreover, viewing the Galileian invariant quantum mechanics as a non-relativistic limit, we prove that the above-mentioned averaging procedure giving rise to Bargmann superselection rule is nothing but an effective de-coherence phenomenon due to time evolution if assuming that real measurements includes a temporal averaging procedure. It happens when the added term Mc 2 is taken in due account in the Hamiltonian operator since, in the dynamical approach, the mass M is an operator and cannot be trivially neglected as in classical mechanics. The presented results are quite general and rely upon the only hypothesis that the mass operator has point spectrum. These results explicitly show the interplay of the period of time of the averaging procedure, the energy content of the considered states, and the minimal difference of the mass operator eigenvalues.
  PubDate: 2013-05-01
Spectral Packing Dimensions through Power-Law Subordinacy
- Abstract: Abstract We offer a method of classification of spectral measures of discrete one-dimensional Schrödinger operators with respect to packing measures, which can be seen as dual to results for Hausdorff measures in subordinacy theory. We apply this method to classes of sparse operators, and give an example whose spectral measure has different Hausdorff and packing dimensions, and others for which such dimensions coincide. Some dynamical motivations are also mentioned.
  PubDate: 2013-05-01
Future Non-Linear Stability for Reflection Symmetric Solutions of the Einstein–Vlasov System of Bianchi Types II and VI₀
- Abstract: Abstract Using the methods developed for the Bianchi I case we have shown that a boostrap argument is also suitable to treat the future non-linear stability for reflection symmetric solutions of the Einstein–Vlasov system of Bianchi types II and VI0. These solutions are asymptotic to the Collins–Stewart solution with dust and the Ellis–MacCallum solution, respectively. We have thus generalized the results obtained by Rendall and Uggla in the case of locally rotationally symmetric Bianchi II spacetimes to the reflection symmetric case. However, we needed to assume small data. For Bianchi VI0 there is no analogous previous result.
  PubDate: 2013-05-01
Absolutely Continuous Spectrum for Random Schrödinger Operators on Tree-Strips of Finite Cone Type
- Abstract: Abstract A tree-strip of finite cone type is the product of a tree of finite cone type with a finite set. We consider random Schrödinger operators on these tree-strips, similar to the Anderson model. We prove that for small disorder, the spectrum is almost surely, purely, absolutely continuous in a certain set.
  PubDate: 2013-05-01
Decoherence for Quantum Markov Semi-Groups on Matrix Algebras
- Abstract: Abstract In this work we describe a necessary and sufficient condition for decoherence of quantum Markov evolutions acting on matrix spaces (according to the definition introduced by Blanchard and Olkiewicz). This condition is related to the spectral analysis of the generator ${\mathcal{L}}$ of the semigroup and is easily stated: the evolution displays decoherence if and only if the maximal algebra ${\mathcal{N}(\mathcal{T})}$ on which the semigroup is *-automorphic contains all the eigenvalues of ${\mathcal{L}}$ associated with eigenvectors with null real part. Moreover, this condition is surely verified when the semigroup admits a faithful invariant state.
  PubDate: 2013-05-01
Instanton Counting and Wall-Crossing for Orbifold Quivers
- Abstract: Abstract Noncommutative Donaldson–Thomas invariants for abelian orbifold singularities can be studied via the enumeration of instanton solutions in a six-dimensional noncommutative ${{\mathcal N}=2}$ gauge theory; this construction is based on the generalized McKay correspondence and identifies the instanton counting with the counting of framed representations of a quiver which is naturally associated with the geometry of the singularity. We extend these constructions to compute BPS partition functions for higher-rank refined and motivic noncommutative Donaldson–Thomas invariants in the Coulomb branch in terms of gauge theory variables and orbifold data. We introduce the notion of virtual instanton quiver associated with the natural symplectic charge lattice which governs the quantum wall-crossing behaviour of BPS states in this context. The McKay correspondence naturally connects our formalism with other approaches to wall-crossing based on quantum monodromy operators and cluster algebras.
  PubDate: 2013-05-01
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: 2013-04-30
Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction
- Abstract: Abstract In this paper we consider lattice systems coupled by local interactions. We prove invariant manifold theorems for whiskered tori (we recall that whiskered tori are quasi-periodic solutions with exponentially contracting and expanding directions in the linearized system). The invariant manifolds we construct generalize the usual (strong) (un)stable manifolds and allow us to consider also non-resonant manifolds. We show that if the whiskered tori are localized near a collection of specific sites, then so are the invariant manifolds. We recall that the existence of localized whiskered tori has recently been proven for symplectic maps and flows in Fontich et al. (J Diff Equ, 2012), but our results do not need that the systems are symplectic. For simplicity we will present first the main results for maps, but we will show that the result for maps imply the results for flows. It is also true that the results for flows can be proved directly following the same ideas.
  PubDate: 2013-04-30
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: 2013-04-28
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: 2013-04-26
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: 2013-04-25
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: 2013-04-23