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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  [2281 journals]
• Static Vacuum Einstein Metrics on Bounded Domains
• Abstract: Abstract We study the existence and uniqueness of solutions to the static vacuum Einstein equations in bounded domains, satisfying the Bartnik boundary conditions of prescribed metric and mean curvature on the boundary.
PubDate: 2015-10-01

• Static Potentials on Asymptotically Flat Manifolds
• Abstract: Abstract We consider the question whether a static potential on an asymptotically flat 3-manifold can have nonempty zero set which extends to the infinity. We prove that this does not occur if the metric is asymptotically Schwarzschild with nonzero mass. If the asymptotic assumption is relaxed to the usual assumption under which the total mass is defined, we prove that the static potential is unique up to scaling unless the manifold is flat. We also provide some discussion concerning the rigidity of complete asymptotically flat 3-manifolds without boundary that admit a static potential.
PubDate: 2015-10-01

• On Self-Adjoint Extensions and Symmetries in Quantum Mechanics
• Abstract: Abstract Given a unitary representation of a Lie group G on a Hilbert space $${\mathcal H}$$ , we develop the theory of G-invariant self-adjoint extensions of symmetric operators using both von Neumann’s theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the G-invariant unbounded operator. We also prove a G-invariant version of the representation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of G-invariant self-adjoint extensions of the Laplace–Beltrami operator on a smooth Riemannian manifold with boundary on which the group G acts. These extensions are labeled by admissible unitaries U acting on the L 2-space at the boundary and having spectral gap at −1. It is shown that if the unitary representation V of the symmetry group G is traceable, then the self-adjoint extension of the Laplace–Beltrami operator determined by U is G-invariant if U and V commute at the boundary. Various significant examples are discussed at the end.
PubDate: 2015-10-01

• Jack Polynomials with Prescribed Symmetry and Some of Their Clustering
Properties
• Abstract: Abstract We study Jack polynomials in N variables, with parameter α, and having a prescribed symmetry with respect to two disjoint subsets of variables. For instance, these polynomials can exhibit a symmetry of type AS, which means that they are antisymmetric in the first m variables and symmetric in the remaining N − m variables. One of our main goals is to extend recent works on symmetric Jack polynomials (Baratta and Forrester in Nucl Phys B 843:362–381, 2011; Berkesch et al. in Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k + 1)-equals ideal, 2013; Bernevig and Haldane in Phys Rev Lett 101:1–4, 2008) and prove that the Jack polynomials with prescribed symmetry also admit clusters of size k and order r, that is, the polynomials vanish to order r when k + 1 variables coincide. We first prove some general properties for generic α, such as their uniqueness as triangular eigenfunctions of operators of Sutherland type, and the existence of their analogues in infinity many variables. We then turn our attention to the case with α = −(k + 1)/(r − 1). We show that for each triplet (k, r, N), there exist admissibility conditions on the indexing sets, called superpartitions, that guaranty both the regularity and the uniqueness of the polynomials. These conditions are also used to establish similar properties for non-symmetric Jack polynomials. As a result, we prove that the Jack polynomials with arbitrary prescribed symmetry, indexed by (k, r, N)-admissible superpartitions, admit clusters of size k = 1 and order r ≥ 2. In the last part of the article, we find necessary and sufficient conditions for the invariance under translation of the Jack polynomials with prescribed symmetry AS. This allows to find special families of superpartitions that imply the existence of clusters of size k > 1 and order r ≥ 2.
PubDate: 2015-10-01

• Locally Covariant Quantum Field Theory with External Sources
• Abstract: Abstract We provide a detailed analysis of the classical and quantized theory of a multiplet of inhomogeneous Klein–Gordon fields, which couple to the spacetime metric and also to an external source term; thus the solutions form an affine space. Following the formulation of affine field theories in terms of presymplectic vector spaces as proposed in Benini et al. (Ann. Henri Poincaré 15:171–211, 2014), we determine the relative Cauchy evolution induced by metric as well as source term perturbations and compute the automorphism group of natural isomorphisms of the presymplectic vector space functor. Two pathological features of this formulation are revealed: the automorphism group contains elements that cannot be interpreted as global gauge transformations of the theory; moreover, the presymplectic formulation does not respect a natural requirement on composition of subsystems. We therefore propose a systematic strategy to improve the original description of affine field theories at the classical and quantized level, first passing to a Poisson algebra description in the classical case. The idea is to consider state spaces on the classical and quantum algebras suggested by the physics of the theory (in the classical case, we use the affine solution space). The state spaces are not separating for the algebras, indicating a redundancy in the description. Removing this redundancy by a quotient, a functorial theory is obtained that is free of the above-mentioned pathologies. These techniques are applicable to general affine field theories and Abelian gauge theories. The resulting quantized theory is shown to be dynamically local.
PubDate: 2015-10-01

• Smooth Non-Zero Rest-Mass Evolution Across Time-Like Infinity
• Abstract: Abstract It is shown that solutions to Einstein’s field equations with positive cosmological constant can include non-zero rest-mass fields which coexist with and travel unimpeded across a smooth conformal boundary. This is exemplified by the coupled Einstein-massive-scalar field equations for which the mass m is related to the cosmological constant λ by the relation 3m 2 = 2 λ. Cauchy data for the conformal field equations can in this case be prescribed on the (compact, space-like) conformal boundary $${\mathcal{J}^{+}}$$ . Their developments backwards in time induce a set of standard Cauchy data on space-like slices for the Einstein-massive-scalar field equations which is open in the set of all Cauchy data for this system.
PubDate: 2015-10-01

• Characteristic Initial Data and Smoothness of Scri. I. Framework and
Results
• Abstract: Abstract We analyze the Cauchy problem for the vacuum Einstein equations with data on a complete light-cone in an asymptotically Minkowskian space-time. We provide conditions on the free initial data which guarantee existence of global solutions of the characteristic constraint equations. We present necessary-and-sufficient conditions on characteristic initial data in 3 + 1 dimensions to have no logarithmic terms in an asymptotic expansion at null infinity.
PubDate: 2015-09-01

• Resonances and Partial Delocalization on the Complete Graph
• Abstract: Abstract Random operators may acquire extended states formed from a multitude of mutually resonating local quasi-modes. This mechanics is explored here in the context of the random Schrödinger operator on the complete graph. The operator exhibits local quasi-modes mixed through a single channel. While most of its spectrum consists of localized eigenfunctions, under appropriate conditions it includes also bands of states which are delocalized in the $${\ell^{1}}$$ -though not in $${\ell^{2}}$$ -sense, where the eigenvalues have the statistics of Šeba spectra. The analysis proceeds through some general observations on the scaling limits of random functions in the Herglotz–Pick class. The results are in agreement with a heuristic condition for the emergence of resonant delocalization, which is stated in terms of the tunneling amplitude among quasi-modes.
PubDate: 2015-09-01

• Conformally Covariant Systems of Wave Equations and their Equivalence to
Einstein’s Field Equations
• Abstract: Abstract We derive, in 3 + 1 spacetime dimensions, two alternative systems of quasi-linear wave equations, based on Friedrich’s conformal field equations. We analyse their equivalence to Einstein’s vacuum field equations when appropriate constraint equations are satisfied by the initial data. As an application, the characteristic initial value problem for the Einstein equations with data on past null infinity is reduced to a characteristic initial value problem for wave equations with data on an ordinary light-cone.
PubDate: 2015-09-01

• Smoothness of Compact Horizons
• Abstract: Abstract We prove that compact Cauchy horizons in a smooth spacetime satisfying the null energy condition are smooth. As an application, we consider the problem of determining when a cobordism admits Lorentzian metrics with certain properties. In particular, we prove a result originally due to Tipler without the smoothness hypothesis necessary in the original proof.
PubDate: 2015-09-01

• Nonequilibrium Markov Processes Conditioned on Large Deviations
• Abstract: Abstract We consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob’s h-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, often used with importance sampling to simulate rare events and giving rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned.
PubDate: 2015-09-01

• The Extension of Distributions on Manifolds, a Microlocal Approach
• Abstract: Abstract Let M be a smooth manifold, $${I\subset M}$$ a closed embedded submanifold of M and U an open subset of M. In this paper, we find conditions using a geometric notion of scaling for $${t\in \mathcal{D}^{\prime}(U{\setminus} I)}$$ to admit an extension in $${\mathcal{D}^\prime(U)}$$ . We give microlocal conditions on t which allow to control the wave front set of the extension generalizing a previous result of Brunetti–Fredenhagen. Furthermore, we show that there is a subspace of extendible distributions for which the wave front of the extension is minimal which has applications for the renormalization of quantum field theory on curved spacetimes.
PubDate: 2015-08-13

• Kauffman Polynomial from a Generalized Yang–Yang Function
• Abstract: Abstract For the fundamental representations of the simple Lie algebras of type B n , C n and D n , we derive the braiding and fusion matrices from the generalized Yang–Yang function and prove that the corresponding knot invariants are Kauffman polynomial.
PubDate: 2015-08-04

• Exponential Decay for the Schrödinger Equation on a Dissipative
Waveguide
• Abstract: Abstract We prove exponential decay for the solution of the Schrödinger equation on a dissipative waveguide. The absorption is effective everywhere on the boundary, but the geometric control condition is not satisfied. The proof relies on separation of variables and the Riesz basis property for the eigenfunctions of the transverse operator. The case where the absorption index takes negative values is also discussed.
PubDate: 2015-08-01

• A Ballistic Motion Disrupted by Quantum Reflections
• Abstract: Abstract I study a Lindblad dynamics modeling a quantum test particle in a Dirac comb that collides with particles from a background gas. The main result is a homogenization theorem in an adiabatic limiting regime involving large initial momentum for the test particle. Over the time interval considered, the particle would exhibit essentially ballistic motion if either the singular periodic potential or the kicks from the gas were removed. However, the particle behaves diffusively when both sources of forcing are present. The conversion of the motion from ballistic to diffusive is generated by occasional quantum reflections that result when the test particle’s momentum is driven through a collision near to an element of the half-spaced reciprocal lattice of the Dirac comb.
PubDate: 2015-08-01

• Gupta–Bleuler Quantization of the Maxwell Field in Globally
Hyperbolic Space-Times
• Abstract: Abstract We give a complete framework for the Gupta–Bleuler quantization of the free electromagnetic field on globally hyperbolic space-times. We describe one-particle structures that give rise to states satisfying the microlocal spectrum condition. The field algebras in the so-called Gupta–Bleuler representations satisfy the time-slice axiom, and the corresponding vacuum states satisfy the microlocal spectrum condition. We also give an explicit construction of ground states on ultrastatic space-times. Unlike previous constructions, our method does not require a spectral gap or the absence of zero modes. The only requirement, the absence of zero-resonance states, is shown to be stable under compact perturbations of topology and metric. Usual deformation arguments based on the time-slice axiom then lead to a construction of Gupta–Bleuler representations on a large class of globally hyperbolic space-times. As usual, the field algebra is represented on an indefinite inner product space, in which the physical states form a positive semi-definite subspace. Gauge transformations are incorporated in such a way that the field can be coupled perturbatively to a Dirac field. Our approach does not require any topological restrictions on the underlying space-time.
PubDate: 2015-08-01

• Poisson Eigenvalue Statistics for Random Schrödinger Operators on
Regular Graphs
• Abstract: Abstract For random operators it is conjectured that spectral properties of an infinite-volume operator are related to the distribution of spectral gaps of finite-volume approximations. In particular, localization and pure point spectrum in infinite volume is expected to correspond to Poisson eigenvalue statistics. Motivated by results on the Anderson model on the infinite tree we consider random Schrödinger operators on finite regular graphs. We study local spectral statistics: we analyze the number of eigenvalues in intervals with length comparable to the inverse of the number of vertices of the graph, in the limit where this number tends to infinity. We show that the random point process generated by the rescaled eigenvalues converges in certain spectral regimes of localization to a Poisson process. The corresponding result on the lattice was proved by Minami. However, due to the geometric structure of regular graphs the known methods turn out to be difficult to adapt. Therefore, we develop a new approach based on direct comparison of eigenvectors.
PubDate: 2015-08-01

• The Multiscale Loop Vertex Expansion
• Abstract: Abstract The loop vertex expansion (LVE) is a constructive technique which uses only canonical combinatorial tools and no space–time dependent lattices. It works for quantum field theories without renormalization. Renormalization requires scale analysis. In this paper, we provide an enlarged formalism which we call the multiscale loop vertex expansion (MLVE). We test it on what is probably the simplest quantum field theory which requires some kind of renormalization, namely a combinatorial model of the vector type with quartic interaction and a propagator which mimicks the power counting of $${\phi^4_2}$$ . An ordinary LVE would fail to treat even this simplest superrenormalizable model, but we show how to perform the ultraviolet limit and prove its analyticity in the Borel summability domain of the model with the MLVE.
PubDate: 2015-08-01

• Torus Knots in Lens Spaces and Topological Strings
• Abstract: Abstract We study the invariant of knots in lens spaces defined from quantum Chern–Simons theory. By means of the knot operator formalism, we derive a generalization of the Rosso-Jones formula for torus knots in L(p,1). In the second part of the paper, we propose a B-model topological string theory description of torus knots in L(2,1).
PubDate: 2015-08-01

• Super-KMS Functionals for Graded-Local Conformal Nets
• Abstract: Abstract Motivated by a few preceding papers and a question of R. Longo, we introduce super-KMS functionals for graded translation-covariant nets over $${\mathbb{R}}$$ with superderivations, roughly speaking as a certain supersymmetric modification of classical KMS states on translation-covariant nets over $${\mathbb{R}}$$ , fundamental objects in chiral algebraic quantum field theory. Although we are able to make a few statements concerning their general structure, most properties will be studied in the setting of specific graded-local (super-) conformal models. In particular, we provide a constructive existence and partial uniqueness proof of super-KMS functionals for the supersymmetric free field, for certain subnets, and for the super-Virasoro net with central charge $${c\ge 3/2}$$ . Moreover, as a separate result, we classify bounded super-KMS functionals for graded-local conformal nets over S 1 with respect to rotations.
PubDate: 2015-08-01

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