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 Annales Henri Poincaré   [SJR: 1.016]   [H-I: 28]   [2 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1424-0637 - ISSN (Online) 1424-0661    Published by Springer-Verlag  [2280 journals]
• Semi-Classical Analysis of Non-Self-Adjoint Transfer Matrices in
Statistical Mechanics I
• Abstract: Abstract We propose a way to study one-dimensional statistical mechanics models with complex-valued action using transfer operators. The argument consists of two steps. First, the contour of integration is deformed so that the associated transfer operator is a perturbation of a normal one. Then the transfer operator is studied using methods of semi-classical analysis. In this paper, we concentrate on the second step, the main technical result being a semi-classical estimate for powers of an integral operator which is approximately normal.
PubDate: 2016-02-01

• On the Validity of the Definition of Angular Momentum in General
Relativity
• Abstract: Abstract We exam the validity of the definition of the ADM angular momentum without the parity assumption. Explicit examples of asymptotically flat hypersurfaces in the Minkowski spacetime with zero ADM energy–momentum vector and finite non-zero angular momentum vector are presented. We also discuss the Beig–Ó Murchadha–Regge–Teitelboim center of mass and study analogous examples in the Schwarzschild spacetime.
PubDate: 2016-02-01

• Resurgent Transseries and the Holomorphic Anomaly
• Abstract: Abstract The gauge theoretic large N expansion yields an asymptotic series which requires a nonperturbative completion to be well defined. Recently, within the context of random matrix models, it was shown how to build resurgent transseries solutions encoding the full nonperturbative information beyond the ’t Hooft genus expansion. On the other hand, via large N duality, random matrix models may be holographically described by B-model closed topological strings in local Calabi–Yau geometries. This raises the question of constructing the corresponding holographically dual resurgent transseries, tantamount to nonperturbative topological string theory. This paper addresses this point by showing how to construct resurgent transseries solutions to the holomorphic anomaly equations. These solutions are built upon (generalized) multi-instanton sectors, where the instanton actions are holomorphic. The asymptotic expansions around the multi-instanton sectors have both holomorphic and anti-holomorphic dependence, may allow for resonance, and their structure is completely fixed by the holomorphic anomaly equations in terms of specific polynomials multiplied by exponential factors and up to the holomorphic ambiguities—which generalizes the known perturbative structure to the full transseries. In particular, the anti-holomorphic dependence has a somewhat universal character. Furthermore, in the non-perturbative sectors, holomorphic ambiguities may be fixed at conifold points. This construction shows the nonperturbative integrability of the holomorphic anomaly equations and sets the ground to start addressing large-order analysis and resurgent nonperturbative completions within closed topological string theory.
PubDate: 2016-02-01

• On the Existence of a Maximal Cauchy Development for the Einstein
Equations: a Dezornification
• Abstract: Abstract In 1969, Choquet-Bruhat and Geroch established the existence of a unique maximal globally hyperbolic Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn’s lemma. In this paper, we present a proof that avoids the use of Zorn’s lemma. In particular, we provide an explicit construction of this maximal globally hyperbolic development.
PubDate: 2016-02-01

• Quantum Phase Transition in an Interacting Fermionic Chain
• Abstract: Abstract We rigorously analyze the quantum phase transition between a metallic and an insulating phase in (non-solvable) interacting spin chains or one-dimensional fermionic systems. In particular, we prove the persistence of Luttinger liquid behavior in the presence of an interaction even arbitrarily close to the critical point, where the Fermi velocity vanishes and the two Fermi points coalesce. The analysis is based on two different multiscale analysis; the analysis of the first regime provides gain factors which compensate the small divisors due to the vanishing Fermi velocity.
PubDate: 2016-02-01

• Dynamical Locality of the Free Maxwell Field
• Abstract: Abstract The extent to which the non-interacting and source-free Maxwell field obeys the condition of dynamical locality is determined in various formulations. Starting from contractible globally hyperbolic spacetimes, we extend the classical field theory to globally hyperbolic spacetimes of arbitrary topology in two ways, obtaining a ‘universal’ theory and a ‘reduced’ theory of the classical free Maxwell field and their corresponding quantisations. We show that the classical and the quantised universal theory fail local covariance and dynamical locality owing to the possibility of having non-trivial radicals in the classical pre-symplectic spaces and non-trivial centres in the quantised *-algebras. The classical and the quantised reduced theory are both locally covariant and dynamically local, thus closing a gap in the discussion of dynamical locality and providing new examples relevant to the question of how theories should be formulated so as to describe the same physics in all spacetimes.
PubDate: 2016-02-01

• Constraint Equations for 3 + 1 Vacuum Einstein Equations with a
Translational Space-Like Killing Field in the Asymptotically Flat Case
• Abstract: Abstract We solve the Einstein constraint equations for a 3 + 1- dimensional vacuum space–time with a space-like translational Killing field. The presence of a space-like translational Killing field allows for a reduction of the 3 + 1-dimensional problem to a 2 + 1-dimensional one. Vacuum Einstein equations with a space-like translational Killing field have been studied by Choquet-Bruhat and Moncrief in the compact case. In the case where an additional rotational symmetry is added, the problem has a long history. In this paper we consider the asymptotically flat case. This corresponds to solving a nonlinear elliptic system on $${\mathbb{R}^2}$$ . The main difficulty in that case is due to the delicate inversion of the Laplacian on $${\mathbb{R}^2}$$ . In particular, we have to work in the non-constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint equations.
PubDate: 2016-02-01

• Long-Time Dynamics of the Perturbed Schrödinger Equation on
Negatively Curved Surfaces
• Abstract: Abstract We consider perturbations of the semiclassical Schrödinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the perturbation, the solutions associated to initial data in a small spectral window become equidistributed in the semiclassical limit. As an application of our method, we also derive some properties of the quantum Loschmidt echo below and beyond the Ehrenfest time for initial data in a small spectral window.
PubDate: 2016-01-30

• On the Spectral Gap of a Quantum Graph
• Abstract: Abstract We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.
PubDate: 2016-01-25

• Abstract Theory of Pointwise Decay with Applications to Wave and
Schrödinger Equations
• Abstract: We prove pointwise in time decay estimates via an abstract conjugate operator method. This is then applied to a large class of dispersive equations.
PubDate: 2016-01-23

• Quantum Hamiltonians with Weak Random Abstract Perturbation. I. Initial
Length Scale Estimate
• Abstract: Abstract We study random Hamiltonians on finite-size cubes and waveguide segments of increasing diameter. The number of random parameters determining the operator is proportional to the volume of the cube. In the asymptotic regime where the cube size, and consequently the number of parameters as well, tends to infinity, we derive deterministic and probabilistic variational bounds on the lowest eigenvalue, i.e., the spectral minimum, as well as exponential off-diagonal decay of the Green function at energies above, but close to the overall spectral bottom.
PubDate: 2016-01-22

• Aharonov–Bohm Effect and High-Momenta Inverse Scattering for the
Klein–Gordon Equation
• Abstract: Abstract We analyze spin-0 relativistic scattering of charged particles propagating in the exterior, $${\Lambda \subset \mathbb{R}^3}$$ , of a compact obstacle $${K \subset \mathbb{R}^3}$$ . The connected components of the obstacle are handlebodies. The particles interact with an electromagnetic field in Λ and an inaccessible magnetic field localized in the interior of the obstacle (through the Aharonov–Bohm effect). We obtain high-momenta estimates, with error bounds, for the scattering operator that we use to recover physical information: we give a reconstruction method for the electric potential and the exterior magnetic field and prove that, if the electric potential vanishes, circulations of the magnetic potential around handles (or equivalently, by Stokes’ theorem, magnetic fluxes over transverse sections of handles) of the obstacle can be recovered, modulo 2π. We additionally give a simple formula for the high momenta limit of the scattering operator in terms of certain magnetic fluxes, in the absence of electric potential. If the electric potential does not vanish, the magnetic fluxes on the handles above referred can be only recovered modulo π and the simple expression of the high-momenta limit of the scattering operator does not hold true.
PubDate: 2016-01-18

• Indefinite Kasparov Modules and Pseudo-Riemannian Manifolds
• Abstract: Abstract We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reversible. We present three examples of our framework: the Dirac operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an indefinite metric); the harmonic oscillator; and the construction via the Kasparov product of an indefinite spectral triple from a family of spectral triples. This last construction corresponds to a foliation of a globally hyperbolic spacetime by spacelike hypersurfaces.
PubDate: 2016-01-18

• The Spectral Properties of the Strongly Coupled Sturm Hamiltonian of
Eventually Constant Type
• Abstract: Abstract We study the spectral properties of the Sturm Hamiltolian of eventually constant type, which includes the Fibonacci Hamiltonian. Let s be the Hausdorff dimension of the spectrum. For V > 20, we show that the restriction of the s-dimensional Hausdorff measure to the spectrum is a Gibbs type measure; the density of states measure is a Markov measure. Based on the fine structures of these measures, we show that both measures are exact dimensional; we obtain exact asymptotic behaviors for the optimal Hölder exponent and the Hausdorff dimension of the density of states measure and for the Hausdorff dimension of the spectrum. As a consequence, if the frequency is not silver number type, then for V big enough, we establish strict inequalities between these three spectral characteristics. We achieve them by introducing an auxiliary symbolic dynamical system and applying the thermodynamical and multifractal formalisms of almost additive potentials.
PubDate: 2016-01-18

• On Supersymmetric Fermion Lattice Systems
• Abstract: Abstract This note provides a C *-algebraic framework for supersymmetry. Particularly, we consider fermion lattice models satisfying the simplest supersymmetry relation. Namely, we discuss a restricted sense of supersymmetry without a boson field involved. We construct general supersymmetric C *-dynamics in terms of a superderivation and a one-parameter group of automorphisms on the CAR algebra. (We do not introduce Grassmann numbers into our formalism.) We show several basic properties of superderivations on the fermion lattice system. Among others, we establish that superderivations defined on the strictly local algebra are norm-closable. We show a criterion of superderivations on the fermion lattice system for being nilpotent. This criterion can be easily checked and hence yields new supersymmetric fermion lattice models.
PubDate: 2016-01-16

• Born–Oppenheimer Approximation for an Atom in Constant Magnetic
Fields
• Abstract: Abstract We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. In Martinez and Sordoni (Mem AMS 936, 2009) such a case is also studied but their reduced Hamiltonian includes the vector potential terms. In this paper, using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms of the nucleus. Using the reduced evolution, we also obtain the asymptotic expansion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constant magnetic fields.
PubDate: 2016-01-14

• Quantum Scattering in a Periodically Pulsed Magnetic Field
• Abstract: Abstract In this paper, we study the quantum dynamics of a charged particle in the plane in the presence of a periodically pulsed magnetic field perpendicular to the plane. We show that by controlling the cycle when the magnetic field is switched on and off appropriately, the result of the asymptotic completeness of wave operators can be obtained under the assumption that the potential V satisfies the decaying condition $${ V(x) \le C(1 + x )^{-\rho}}$$ for some $${\rho > 0}$$ .
PubDate: 2016-01-12

• Construction of Real-Valued Localized Composite Wannier Functions for
Insulators
• Abstract: Abstract We consider a real periodic Schrödinger operator and a physically relevant family of $${m \geq 1}$$ Bloch bands, separated by a gap from the rest of the spectrum, and we investigate the localization properties of the corresponding composite Wannier functions. To this aim, we show that in dimension $${d \leq 3}$$ , there exists a global frame consisting of smooth quasi-Bloch functions which are both periodic and time-reversal symmetric. Aiming to applications in computational physics, we provide a constructive algorithm to obtain such a Bloch frame. The construction yields the existence of a basis of composite Wannier functions which are real-valued and almost-exponentially localized. The proof of the main result exploits only the fundamental symmetries of the projector on the relevant bands, allowing applications, beyond the model specified above, to a broad range of gapped periodic quantum systems with a time-reversal symmetry of bosonic type.
PubDate: 2016-01-01

• Spectral Flows Associated to Flux Tubes
• Abstract: Abstract When a flux quantum is pushed through a gapped two- dimensional tight-binding operator, there is an associated spectral flow through the gap which is shown to be equal to the index of a Fredholm operator encoding the topology of the Fermi projection. This is a natural mathematical formulation of Laughlin’s Gedankenexperiment. It is used to provide yet another proof of the bulk-edge correspondence. Furthermore, when applied to systems with time reversal symmetry, the spectral flow has a characteristic $${\mathbb{Z}_2}$$ signature, while for particle–hole symmetric systems it leads to a criterion for the existence of zero energy modes attached to half-flux tubes. Combined with other results, this allows to explain all strong invariants of two-dimensional topological insulators in terms of a single Fredholm operator.
PubDate: 2016-01-01

• Erratum to: Integrable QFT and Longo–Witten Endomorphisms
• PubDate: 2015-11-25

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