ALBERT

Description:    A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated by the sign-indefinite single-site potential, which is however sign-definite at the boundary of its support. For this class of Anderson operators, we establish a finite-volume criterion which implies that the fractional moment decay property holds. This constructive criterion is satisfied at typical perturbative regimes, e.g. at spectral boundaries which satisfy “Lifshitz tail estimates” on the density of states and for sufficiently strong disorder. We also show how the fractional moment method facilitates the proof of exponential (spectral) localization for such random potentials. Content Type Journal Article Pages 1-29 DOI 10.1007/s00023-011-0112-5 Authors Alexander Elgart, 448 Department of Mathematics, McBryde Hall, Virginia Tech, Blacksburg, VA 24061, USA Martin Tautenhahn, Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Ivan Veselić, Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We place further restriction on the possible topology of stationary asymptotically flat vacuum black holes in five spacetime dimensions. We prove that the horizon manifold can be either a connected sum of Lens spaces and “handles” S 1 × S 2 , or the quotient of S 3 by certain finite groups of isometries (with no “handles”). The resulting horizon topologies include Prism manifolds and quotients of the Poincare homology sphere. We also show that the topology of the domain of outer communication is a cartesian product of the time direction with a finite connected sum of \mathbb R 4 , S 2 × S 2 ’s and CP 2 ’s, minus the black hole itself. We do not assume the existence of any Killing vector beside the asymptotically time like one required by definition for stationarity. Content Type Journal Article Pages 1-23 DOI 10.1007/s00023-011-0079-2 Authors Stefan Hollands, School of Mathematics, Cardiff University, Cardiff, UK Jan Holland, School of Mathematics, Cardiff University, Cardiff, UK Akihiro Ishibashi, KEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), Tsukuba, Japan Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In this paper, we perform the 1/ N expansion of the colored three-dimensional Boulatov tensor model. As in matrix models, we obtain a systematic topological expansion, with increasingly complicated topologies suppressed by higher and higher powers of N . We compute the first orders of the expansion and prove that only graphs corresponding to three spheres S 3 contribute to the leading order in the large N limit. Content Type Journal Article Pages 1-19 DOI 10.1007/s00023-011-0101-8 Authors Razvan Gurau, Perimeter Institute for Theoretical Physics, 31 Caroline St., Waterloo, Ontario N2L 2Y5, Canada Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Ringström managed (in Invent Math 173(1):123–208, 2008 ) to prove future stability of solutions to Einstein’s field equations when matter consists of a scalar field with a potential creating an accelerated expansion. This was done for a quite wide class of spatially homogeneous space–times. The methods he used should be applicable also when other kinds of matter fields are added to the stress-energy tensor. This article addresses the question whether we can obtain stability results similar to those Ringström obtained if we add an electromagnetic field to the matter content. Before this question can be addressed, more general properties concerning Einstein’s field equation coupled to a scalar field and an electromagnetic field have to be settled. The most important of these questions are the existence of a maximal globally hyperbolic development and the Cauchy stability of solutions to the initial value problem. Content Type Journal Article Pages 1-69 DOI 10.1007/s00023-011-0099-y Authors Christopher Svedberg, Department of Mathematics, KTH, Lindstedtsvägen 25, 100 44 Stockholm, Sweden Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    This paper contains the second part of a two-part series on the stability and instability of extreme Reissner–Nordström spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation \square g y = 0 on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ 0 crossing the future event horizon H + . We here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon H + . A hierarchy of conservations laws on degenerate horizons is also derived. Content Type Journal Article Pages 1-48 DOI 10.1007/s00023-011-0110-7 Authors Stefanos Aretakis, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB UK Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    It is illustrated by a class of counter examples why the Brill–Cantor criterion is not sufficient to ensure the solvability of the Lichnerowicz equation for asymptotically flat, time reflection symmetric-free data. Content Type Journal Article Pages 1-7 DOI 10.1007/s00023-011-0102-7 Authors Helmut Friedrich, Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, 14476 Golm, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We study the low-energy asymptotics of the spectral shift function for Schrödinger operators with potentials decaying like O (\frac1 | x | 2 ) . We prove a generalized Levinson’s theorem for this class of potentials in presence of zero eigenvalue and zero resonance. Content Type Journal Article Pages 1-44 DOI 10.1007/s00023-011-0117-0 Authors Xiaoyao Jia, Department of Mathematics, Henan University of Sciences and Technology, Luoyang, 471003 China François Nicoleau, Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, Université de Nantes, 44322 Nantes Cedex 3, France Xue Ping Wang, Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, Université de Nantes, 44322 Nantes Cedex 3, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In this work we give a positive answer to the following question: does Stochastic Mechanics uniquely define a three-dimensional stochastic process which describes the motion of a particle in a Bose–Einstein condensate? To this extent we study a system of N trapped bosons with pair interaction at zero temperature under the Gross–Pitaevskii scaling, which allows to give a theoretical proof of Bose–Einstein condensation for interacting trapped gases in the limit of N going to infinity. We show that under the assumption of strictly positivity and continuous differentiability of the many-body ground state wave function it is possible to rigorously define a one-particle stochastic process, unique in law, which describes the motion of a single particle in the gas and we show that, in the scaling limit, the one-particle process continuously remains outside a time dependent random “interaction-set” with probability one. Moreover, we prove that its stopped version converges, in a relative entropy sense, toward a Markov diffusion whose drift is uniquely determined by the order parameter, that is the wave function of the condensate. Content Type Journal Article Pages 1-12 DOI 10.1007/s00023-011-0116-1 Authors Laura M. Morato, Facoltà di Scienze, Università di Verona, Strada le Grazie, 37134 Verona, Italy Stefania Ugolini, Dipartimento di Matematica, Università di Milano, via Saldini, Milan, Italy Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider a dynamic mean-field ferromagnetic model in the low-temperature regime in the neighborhood of the zero magnetization state. We study the random time it takes for the system to make a decision, i.e., to exit the neighborhood of the unstable equilibrium and approach one of the two stable equilibrium points. We prove a limit theorem for the distribution of this random time in the thermodynamic limit. Content Type Journal Article Pages 1-13 DOI 10.1007/s00023-011-0148-6 Authors Yuri Bakhtin, School of Mathematics, Georgia Tech, Atlanta, GA 30332-0160, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider the semiclassical asymptotics of the sum of negative eigenvalues of the three-dimensional Pauli operator with an external potential and a self-generated magnetic field B . We also add the field energy b ó õ B 2 and we minimize over all magnetic fields. The parameter β effectively determines the strength of the field. We consider the weak field regime with β h 2 ≥ const  〉 0, where h is the semiclassical parameter. For smooth potentials we prove that the semiclassical asymptotics of the total energy is given by the non-magnetic Weyl term to leading order with an error bound that is smaller by a factor h 1+ e , i.e. the subleading term vanishes. However for potentials with a Coulomb singularity, the subleading term does not vanish due to the non-semiclassical effect of the singularity. Combined with a multiscale technique, this refined estimate is used in the companion paper (Erdős et al. in Scott correction for large molecules with a self-generated magnetic field, Preprint, 2011 ) to prove the second order Scott correction to the ground state energy of large atoms and molecules. Content Type Journal Article Pages 1-60 DOI 10.1007/s00023-011-0150-z Authors László Erdős, Institute of Mathematics, University of Munich, Theresienstr. 39, 80333 Munich, Germany Søren Fournais, Department of Mathematical Sciences, Aarhus University, Ny Munkegade 118, 8000 Aarhus, Denmark Jan Philip Solovej, Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We prove persistence of absolutely continuous spectrum for the Anderson model on a general class of tree-like graphs. Content Type Journal Article Pages 1-23 DOI 10.1007/s00023-011-0139-7 Authors Florina Halasan, Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover, in the same spirit as the notion of KAM stable integrable Hamiltonians, we will introduce a notion of effectively stable integrable Hamiltonians, conjecture a characterization of these Hamiltonians and show that our result proves this conjecture in the linear case. Content Type Journal Article Pages 1-12 DOI 10.1007/s00023-011-0137-9 Authors Abed Bounemoura, IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320 Brazil Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We study the scattering theory for charged Klein–Gordon equations: { lll ( ¶ t - i v ( x )) 2 f ( t , x )+ e 2 ( x , D x ) f ( t , x )=0, f (0, x ) = f 0 , i - 1 ¶ t f (0, x ) = f 1 , where lll e 2 ( x , D x ) = - å 1 £ j , k £ n   ( ¶ x j - i b j ( x )) a jk ( x )( ¶ x k - i b k ( x ))+ m 2 ( x ), describing a Klein–Gordon field minimally coupled to an external electromagnetic field described by the electric potential v ( x ) and magnetic potential ® b   ( x ) . The flow of the Klein–Gordon equation preserves the energy: lll h [ f , f ]: = ó õ \mathbb R n   f   1   ( x ) f 1 ( x )+ f   0   ( x ) e 2 ( x , D x ) f 0 ( x ) - f   0   ( x ) v 2 ( x ) f 0 ( x ) d x . We consider the situation when the energy is not positive. In this case the flow cannot be written as a unitary group on a Hilbert space, and the Klein–Gordon equation may have complex eigenfrequencies. Using the theory of definitizable operators on Krein spaces and time-dependent methods, we prove the existence and completeness of wave operators, both in the short- and long-range cases. The range of the wave operators are characterized in terms of the spectral theory of the generator, as in the usual Hilbert space case. Content Type Journal Article Pages 1-59 DOI 10.1007/s00023-011-0138-8 Authors Christian Gérard, Département de Mathématiques, Université de Paris XI, 91405 Orsay Cedex, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We study the properties of the ergosurface of the Pomeransky–Senkov black rings, and show that it splits into an “inner” and an “outer” region. As for the singular set, the topology of the “outer ergosurface” depends upon the value of parameters. Content Type Journal Article Pages 1-14 DOI 10.1007/s00023-011-0149-5 Authors Julien Cortier, Institut de Mathématiques et de Modélisation de Montpellier, Université Montpellier 2, CC0051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We analyze a general class of difference operators H e = T e + V e on l 2 (( e \mathbb Z ) d ) where V e is a multi-well potential and e is a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we shall treat the eigenvalue problem for H e as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix, similar to the analysis for the Schrödinger operator [see Helffer and Sjöstrand in Commun Partial Differ Equ 9:337–408, 1984 ], and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix. Content Type Journal Article Pages 1-39 DOI 10.1007/s00023-011-0152-x Authors Markus Klein, Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany Elke Rosenberger, Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We generalize key aspects of gr-qc/1010.5367 (and also gr-qc/1010.5327) to the case of massless l f 2 n quantum field theory on deSitter spacetime. As in that paper, our key objective is to derive a suitable “Mellin-Barnes-type” representation of deSitter correlation functions in a deSitter-invariant state, which holds to arbitrary orders in perturbation theory, and which incorporates renormalization. The representation is suitable for the study of large distance/time properties of correlation functions. It is arrived at via an analytic continuation from the corresponding objects on the sphere, and, as in the massive case, relies on the use of graph-polynomials and their properties, as well as other tools. However, the perturbation expansion is organized somewhat differently in the massless case, due to the well-known subtleties associated with the “zero-mode” of the quantum field. In particular, the correlation functions do not possess a well-defined limit as the self-coupling constant of the field goes to zero, reflecting the well-known non-existence of a deSitter invariant state in the free massless scalar theory. We establish that generic correlation functions cannot grow more than polynomially in proper time for large time-like separations of the points. Our results thus leave open the possibility of quantum-induced IR-instabilities of deSitter spacetime on very large time-scales. Content Type Journal Article Pages 1-43 DOI 10.1007/s00023-011-0140-1 Authors Stefan Hollands, School of Mathematics, Cardiff University, Cardiff, Wales, UK Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We study infrared divergences due to ultraviolet–infrared mixing in quantum field theory on Moyal space with Lorentzian signature in the Yang–Feldman formalism. Concretely, we are considering the f 4 and the f 3 model in arbitrary even dimension. It turns out that the situation is worse than in the Euclidean setting, in the sense that we find infrared divergences in graphs that are finite there. We briefly discuss the problems one faces when trying to adapt the nonlocal counterterms that render the Euclidean model renormalizable. Content Type Journal Article Pages 1-19 DOI 10.1007/s00023-011-0153-9 Authors Jochen Zahn, Courant Research Centre “Higher Order Structures”, Universityof Göttingen, Bunsenstraße 3-5, 37073 Göttingen, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    The sum of the first n ≥ 1 energy levels of the planar Laplacian with constant magnetic field of given total flux is shown to be maximal among triangles for the equilateral triangle, under normalization of the ratio (moment of inertia)/(area) 3 on the domain. The result holds for both Dirichlet and Neumann boundary conditions, with an analogue for Robin (or de Gennes) boundary conditions too. The square similarly maximizes the eigenvalue sum among parallelograms, and the disk maximizes among ellipses. More generally, a domain with rotational symmetry will maximize the magnetic eigenvalue sum among all linear images of that domain. These results are new even for the ground state energy ( n  = 1). Content Type Journal Article Pages 1-20 DOI 10.1007/s00023-011-0142-z Authors Richard S. Laugesen, Department of Mathematics, University of Illinois, Urbana, IL 61801, USA Jian Liang, Department of Mathematics, University of Illinois, Urbana, IL 61801, USA Arindam Roy, Department of Mathematics, University of Illinois, Urbana, IL 61801, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Consider a small sample coupled to a finite number of leads and assume that the total (continuous) system is at thermal equilibrium in the remote past. We construct a non-equilibrium steady state (NESS) by adiabatically turning on an electrical bias between the leads. The main mathematical challenge is to show that certain adiabatic wave operators exist and to identify their strong limit when the adiabatic parameter tends to zero. Our NESS is different from, though closely related with the NESS provided by the Jakšić–Pillet–Ruelle approach. Thus we partly settle a question asked by Caroli et al. (J. Phys. C Solid State Phys. 4(8):916–929, 1971 ) regarding the (non)equivalence between the partitioned and partition-free approaches. Content Type Journal Article Pages 1-30 DOI 10.1007/s00023-011-0144-x Authors Horia D. Cornean, Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg, Denmark Pierre Duclos, Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg, Denmark Radu Purice, Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg, Denmark Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We study discrete Schrödinger operators with compactly supported potentials on Z d . Constructing spectral representations and representing S-matrices by the generalized eigenfunctions, we show that the potential is uniquely reconstructed from the S-matrix of all energies. We also study the spectral shift function x ( l ) for the trace class potentials, and estimate the discrete spectrum in terms of the moments of x ( l ) and the potential. Content Type Journal Article Pages 1-38 DOI 10.1007/s00023-011-0141-0 Authors Hiroshi Isozaki, Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571 Japan Evgeny Korotyaev, Mathematical Physics Department, Faculty of Physics, Petersburg State University, St. Ulianovskaya 2, St. Petersburg, 198904 Russia Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In this paper, we show that massless Dirac waves in the Schwarzschild geometry decay to zero at a rate t −2 λ , where λ = 1, 2, . . . is the angular momentum. Our technique is to use Chandrasekhar’s separation of variables whereby the Dirac equations split into two sets of wave equations. For the first set, we show that the wave decays as t −2 λ . For the second set, in general, the solutions tend to some explicit profile at the rate t −2 λ . The decay rate of solutions of Dirac equations is achieved by showing that the coefficient of the explicit profile is exactly zero. The key ingredients in the proof of the decay rate of solutions for the first set of wave equations are an energy estimate used to show the absence of bound states and zero energy resonance and the analysis of the spectral representation of the solutions. The proof of asymptotic behavior for the solutions of the second set of wave equations relies on careful analysis of the Green’s functions for time independent Schrödinger equations associated with these wave equations. Content Type Journal Article Pages 1-47 DOI 10.1007/s00023-011-0145-9 Authors Joel Smoller, Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109, USA Chunjing Xie, Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We characterize the crossover regime to the KPZ equation for a class of one-dimensional weakly asymmetric exclusion processes. The crossover depends on the strength asymmetry an 2- γ ( a , γ 〉  0) and it occurs at γ  = 1/2. We show that the density field is a solution of an Ornstein-Uhlenbeck equation if g Î (1/2,1] , while for γ  = 1/2 it is an energy solution of the KPZ equation. The corresponding crossover for the current of particles is readily obtained. Content Type Journal Article Pages 1-14 DOI 10.1007/s00023-011-0147-7 Authors Patrícia Gonçalves, CMAT, Centro de Matemática da Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal Milton Jara, IMPA, Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, Brazil Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We initiate the study of the spherically symmetric Einstein–Klein–Gordon system in the presence of a negative cosmological constant, a model appearing frequently in the context of high-energy physics. Due to the lack of global hyperbolicity of the solutions, the natural formulation of dynamics is that of an initial boundary value problem, with boundary conditions imposed at null infinity. We prove a local well-posedness statement for this system, with the time of existence of the solutions depending only on an invariant H 2 -type norm measuring the size of the Klein–Gordon field on the initial data. The proof requires the introduction of a renormalized system of equations and relies crucially on r -weighted estimates for the wave equation on asymptotically AdS spacetimes. The results provide the basis for our companion paper establishing the global asymptotic stability of Schwarzschild-Anti-de-Sitter within this system. Content Type Journal Article Pages 1-48 DOI 10.1007/s00023-011-0146-8 Authors Gustav Holzegel, Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA Jacques Smulevici, Max-Planck-Institut für Gravitationsphysik, Albert-Einstein Institut, Am Mühlenberg 1, 14476 Golm, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    The 6j-symbol is a fundamental object from the re-coupling theory of SU(2) representations. In the limit of large angular momenta, its asymptotics is known to be described by the geometry of a tetrahedron with quantized lengths. This article presents a new recursion formula for the square of the 6j-symbol. In the asymptotic regime, the new recursion is shown to characterize the closure of the relevant tetrahedron. Since the 6j-symbol is the basic building block of the Ponzano–Regge model for pure three-dimensional quantum gravity, we also discuss how to generalize the method to derive more general recursion relations on the full amplitudes. Content Type Journal Article Pages 1-17 DOI 10.1007/s00023-011-0143-y Authors Valentin Bonzom, Perimeter Institute, 31 Caroline St. N, Waterloo, ON N2L 2Y5, Canada Etera R. Livine, Perimeter Institute, 31 Caroline St. N, Waterloo, ON N2L 2Y5, Canada Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Motivated by recent work of Choquet-Bruhat et al. (Class Quantum Gravity 26(135011), 22, 2009 ), we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop–Gromov relative volume monotonicity theorem and Günther’s volume comparison theorem. We briefly discuss how these estimates may be used to control the null second fundamental form of slices of the null cone in Ricci-flat Lorentzian four-manifolds with null curvature bounded above. Content Type Journal Article Pages 1-21 DOI 10.1007/s00023-011-0090-7 Authors James D. E. Grant, Institut für Grundlagen der Bauingenieurwissenschaften, Leopold-Franzens-Universität Innsbruck, Technikerstrasse 13, 6020 Innsbruck, Austria Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider the operator associated with a random walk on finite volume surfaces with hyperbolic cusps. We study the spectral gap (upper and lower bound) associated with this operator and deduce some rate of convergence of the iterated kernel towards its stationary distribution. Content Type Journal Article Pages 1-33 DOI 10.1007/s00023-011-0085-4 Authors Hans Christianson, Department of Mathematics, UNC-Chapel Hill, CB#3250 Phillips Hall, Chapel Hill, NC 27599, USA Colin Guillarmou, DMA, U.M.R. 8553, CNRS, Ecole Normale Supérieure, 45, rue d’Ulm, 75230 Paris Cedex 05, France Laurent Michel, Lab. Dieudonné, Univ. de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We prove that the Hamiltonian of the model describing a spin which is linearly coupled to a field of relativistic and massless bosons, also known as the spin-boson model, admits a ground state for small values of the coupling constant λ. We show that the ground-state energy is an analytic function of λ and that the corresponding ground state can also be chosen to be an analytic function of λ. No infrared regularization is imposed. Our proof is based on a modified version of the BFS operator theoretic renormalization analysis. Moreover, using a positivity argument we prove that the ground state of the spin-boson model is unique. We show that the expansion coefficients of the ground state and the ground-state energy can be calculated using regular analytic perturbation theory. Content Type Journal Article Pages 1-57 DOI 10.1007/s00023-011-0091-6 Authors David Hasler, Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA Ira Herbst, Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In this paper, we prove the following theorem regarding the Wang–Yau quasi-local energy of a spacelike two-surface in a spacetime: Let Σ be a boundary component of some compact, time-symmetric, spacelike hypersurface Ω in a time-oriented spacetime N satisfying the dominant energy condition. Suppose the induced metric on Σ has positive Gaussian curvature and all boundary components of Ω have positive mean curvature. Suppose H ≤ H 0 where H is the mean curvature of Σ in Ω and H 0 is the mean curvature of Σ when isometrically embedded in \mathbb R 3 . If Ω is not isometric to a domain in \mathbb R 3 , then 1.  the Brown–York mass of Σ in Ω is a strict local minimum of the Wang–Yau quasi-local energy of Σ. 2.  on a small perturbation ~ S   of Σ in N , there exists a critical point of the Wang–Yau quasi-local energy of ~ S   . Content Type Journal Article Pages 1-31 DOI 10.1007/s00023-011-0097-0 Authors Pengzi Miao, School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia Luen-Fai Tam, Department of Mathematics, The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, Hong Kong, China Naqing Xie, School of Mathematical Sciences, Fudan University, Shanghai, 200433 China Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Vector potentials are known to have a direct significance to quantum particles moving in the magnetic field. This is called the Aharonov–Bohm effect and is known as one of the most remarkable quantum phenomena. Here we study this quantum effect through the resonance problem. We consider the scattering system consisting of two scalar potentials and one magnetic field with supports at large separation in two dimensions. The system has trajectories oscillating between these supports. We give a sharp lower bound on the resonance widths as the distances between the three supports go to infinity. The bound is described in terms of the backward amplitude for scattering by each of the scalar potentials and by the magnetic field, and it also depends heavily on the magnetic flux of the field. Content Type Journal Article Pages 1-43 DOI 10.1007/s00023-011-0088-1 Authors Ivana Alexandrova, Department of Mathematics and Statistics, State University of New York, 1400 Washington Avenue, ES 110, Albany, NY 12222, USA Hideo Tamura, Department of Mathematics, Okayama University, Okayama, 700-8530 Japan Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We analyze the spectrum of many-body Coulomb Hamiltonians in the setting of open Riemannian manifolds, proving stability of matter in any complete noncompact Riemannian three-manifold of nonnegative Ricci curvature and Euclidean volume growth. Content Type Journal Article Pages 1-19 DOI 10.1007/s00023-011-0084-5 Authors Alberto Enciso, Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UC3M), Consejo Superior de Investigaciones Cientí ficas, 28049 Madrid, Spain Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    This paper is devoted to the study of dimension theory, in particular multifractal analysis, for multimodal maps. We describe the Lyapunov spectrum and study the multifractal spectrum of pointwise dimension. The lack of regularity of the thermodynamic formalism for this class of maps is reflected in the phase transitions of the spectra. Content Type Journal Article Pages 1-30 DOI 10.1007/s00023-011-0086-3 Authors Godofredo Iommi, Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Avenida Vicuña Mackenna, 4860 Santiago, Chile Mike Todd, Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Quantum field theory on the noncommutative two-dimensional Minkowski space with Grosse–Wulkenhaar potential is discussed in two ways: in terms of a continuous set of generalised eigenfunctions of the wave operator, and directly in position space. In both settings, we find a new type of divergence in planar graphs. It is present at and above the self-dual point. This new kind of divergence might make the construction of a Minkowski space version of the Grosse–Wulkenhaar model impossible. Content Type Journal Article Pages 1-28 DOI 10.1007/s00023-011-0089-0 Authors Jochen Zahn, Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In this paper the existence of a class of self-similar solutions of the Einstein–Vlasov system is proved. The initial data for these solutions are not smooth, with their particle density being supported in a submanifold of codimension one. They can be thought of as intermediate between smooth solutions of the Einstein–Vlasov system and dust. The motivation for studying them is to obtain insights into possible violation of weak cosmic censorship by solutions of the Einstein–Vlasov system. By assuming a suitable form of the unknowns it is shown that the existence question can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters. This solution starts at a particular point P 0 and converges to a stationary solution P 1 as the independent variable tends to infinity. The existence proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes. The spacetimes constructed do not constitute a counterexample to cosmic censorship since they are not asymptotically flat. They should be seen as the first step on a possible path towards constructing solutions of importance for understanding the issue of the formation of naked singularities in general relativity. Content Type Journal Article Pages 1-46 DOI 10.1007/s00023-011-0094-3 Authors Alan D. Rendall, Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, 14476 Potsdam, Germany Juan J. L. Velázquez, ICMAT (CSIC-UAM-UC3M-UCM), C/ Nicolás Cabrera 15, Cantoblanco, 28049 Madrid Spain Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Let \mathfrak g be a simple Lie algebra, L \mathfrak g be the loop algebra of \mathfrak g . Fixing a point in S 1 and identifying the real line with the punctured circle, we consider the subalgebra \fancyscript S \mathfrak g of L \mathfrak g of rapidly decreasing elements on \mathbb R . We classify the translation-invariant 2-cocycles on \fancyscript S \mathfrak g . We show that the ground state representation of \fancyscript S \mathfrak g is unique for each cocycle. These ground states correspond precisely to the vacuum representations of L \mathfrak g . Content Type Journal Article Pages 1-23 DOI 10.1007/s00023-011-0095-2 Authors Yoh Tanimoto, Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133 Rome, Italy Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs. Content Type Journal Article Pages 1-27 DOI 10.1007/s00023-011-0098-z Authors Jean Bourgain, School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA Zeév Rudnick, Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    The recent “breakdown criterion” result of Klainerman and Rodnianski (On the breakdown criterion in general relativity, Preprint, 2009 ) stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can be further extended in time if the second fundamental form and the derivative of the lapse of the foliation are uniformly bounded. This theorem and its proof were extended to Einstein-scalar and Einstein–Maxwell spacetimes in the thesis (Shao in Breakdown criteria for nonvacuum Einstein equations, Ph.D. thesis, Princeton University, 2010 ). In this paper, we state the main results of Shao ( 2010 ), and we summarize and discuss their proofs. In particular, we will discuss the various issues resulting from nontrivial Ricci curvature and the coupling between the Einstein and the field equations. Content Type Journal Article Pages 1-73 DOI 10.1007/s00023-011-0082-7 Authors Arick Shao, Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We determine the orbit types of the action of the group of local gauge transformations on the space of connections in a principal bundle with structure group O( n ), SO( n ) or Sp( n ) over a closed, simply connected manifold of dimension 4. On the way we derive a classification of Howe subgroups of SO( n ) up to conjugacy. Content Type Journal Article Pages 1-45 DOI 10.1007/s00023-011-0081-8 Authors Alexander Hertsch, Institut für Theoretische Physik, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany Gerd Rudolph, Institut für Theoretische Physik, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany Matthias Schmidt, Institut für Theoretische Physik, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary space–time dimensions n  + 1 ≥ 3. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case. Content Type Journal Article Pages 1-64 DOI 10.1007/s00023-011-0076-5 Authors Yvonne Choquet-Bruhat, Académie des Sciences, 23, quai de Conti, 75006 Paris, France Piotr T. Chruściel, Gravitational Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria José M. Martín-García, Institut d’Astrophysique de Paris and Laboratoire Univers et Théories, UMR 8102, Observatoire de Paris-Meudon, 5, place Jules Janssen, 92195 Meudon, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description: Résumé   Nous considérons une classe assez générale de systèmes différentiels sur le cercle avec une perturbation aléatoire d’ordre inférieur. Nous adoptons deux points de vue, semiclassique et haute fréquence. Nous montrons (a) que dans la limite h → 0, les valeurs propres se distribuent selon une loi de Weyl avec une probabilité très proche de 1, (b) que les grandes valeurs propres se distribuent presque sûrement selon une loi de Weyl. Content Type Journal Article Pages 1-32 DOI 10.1007/s00023-010-0073-0 Authors William Bordeaux Montrieux, Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, 91120 Palaiseau Cedex, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We prove the existence of a continuous family of finite-energy particle-like solutions in the coupled Georgi–Glashow–Skyrme model carrying both electric and magnetic charges, known as dyons. Due to the presence of electricity and the Minkowski spacetime signature, we need to solve a variational problem with an indefinite action functional. Our results show that, while the magnetic charge is uniquely determined by the topological monopole number, the electric charge of a solution can be arbitrarily prescribed in an open interval. Content Type Journal Article Pages 1-21 DOI 10.1007/s00023-011-0083-6 Authors Fanghua Lin, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA Yisong Yang, Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, NY 11201, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Explicit Fermi coordinates are given for geodesic observers comoving with the Hubble flow in expanding Robertson–Walker space–times, along with exact expressions for the metric tensors in Fermi coordinates. For the case of non-inflationary cosmologies, it is shown that Fermi coordinate charts are global, and space–time is foliated by space slices of constant Fermi (proper) time that have finite extent. A universal upper bound for the proper radius of any leaf of the foliation, i.e., for the proper radius of the spatial universe at any fixed time of the geodesic observer, is given. A general expression is derived for the geometrically defined Fermi relative velocity of a test particle (e.g., a galaxy cluster) comoving with the Hubble flow away from the observer. Least upper bounds of superluminal recessional Fermi velocities are given for space–times whose scale factors follow power laws, including matter-dominated and radiation-dominated cosmologies. Exact expressions for the proper radius of any leaf of the foliation for this same class of space–times are given. It is shown that the radii increase linearly with proper time of the observer moving with the Hubble flow. These results are applied to particular cosmological models. Content Type Journal Article Pages 1-26 DOI 10.1007/s00023-011-0080-9 Authors David Klein, Department of Mathematics and Interdisciplinary, Research Institute for the Sciences, California State University Northridge, Northridge, CA 91330-8313, USA Evan Randles, Department of Mathematics, California State University Northridge, Northridge, CA 91330-8313, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We systematically describe and classify one-dimensional Schrödinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe two new classes of exactly solvable Schrödinger equations that can be reduced to the Hermite equation. Content Type Journal Article Pages 1-22 DOI 10.1007/s00023-011-0077-4 Authors Jan Dereziński, Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoża 74, 00-682 Warsaw, Poland Michał Wrochna, RTG “Mathematical Structures in Modern Quantum Physics”, Institute of Mathematics, University of Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    The differential expression L m = - ¶ x 2 +( m 2 - 1/4) x - 2 defines a self-adjoint operator H m on L 2 (0, ∞) in a natural way when m 2  ≥ 1. We study the dependence of H m on the parameter m show that it has a unique holomorphic extension to the half-plane Re m 〉 −1, and analyze spectral and scattering properties of this family of operators. Content Type Journal Article Pages 1-44 DOI 10.1007/s00023-011-0078-3 Authors Laurent Bruneau, Department of Mathematics and UMR 8088 CNRS, University of Cergy-Pontoise, 95000 Cergy-Pontoise, France Jan Dereziński, Department of Mathematical Methods in PhysicsFaculty of Physics, University of Warsaw, Hoża 74, 00-682 Warsaw, Poland Vladimir Georgescu, CNRS and University of Cergy-Pontoise, 95000 Cergy-Pontoise, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In this paper, we employ a technique combining the Euler Maclaurin formula with the saddle point approximation method to obtain the asymptotic behavior (in the limit of large representation index J ) of generic Wigner matrix elements D J MM ¢ ( g ) . We use this result to derive asymptotic formulae for the character χ J ( g ) of an SU (2) group element and for Wigner’s 3 j symbol. Surprisingly, given that we perform five successive layers of approximations, the asymptotic formula we obtain for χ J ( g ) is in fact exact . The result hints at a “Duistermaat-Heckman like” localization property for discrete sums. Content Type Journal Article Pages 1-42 DOI 10.1007/s00023-010-0072-1 Authors Joseph Ben Geloun, Perimeter Institute for Theoretical Physics, 31, Caroline N., Waterloo, ON N2L 2Y5, Canada Razvan Gurau, Perimeter Institute for Theoretical Physics, 31, Caroline N., Waterloo, ON N2L 2Y5, Canada Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Let B  = ( B 1 ( t ), . . . , B d ( t )) be a d -dimensional fractional Brownian motion with Hurst index α  〈 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B , or to solving differential equations driven by B . We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using “standard” tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates and call for an extension of Gaussian tools such as, for instance, the Malliavin calculus. After a first introductory paper (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics, 2011 ), this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as Lévy area. A summary in French may be found in Unterberger (Mode d’emploi de la théorie constructive des champs bosoniques, avec une application aux chemins rugueux, 2011 ). Content Type Journal Article Pages 1-62 DOI 10.1007/s00023-011-0119-y Authors Jacques Magnen, CPHT, Ecole Polytechnique, CNRS, UMR 7644, 91128 Palaiseau Cedex, France Jérémie Unterberger, Université Henri Poincaré, BP 239, 54506 Vandoeuvre-les-Nancy Cedex, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    The present article considers time-symmetric initial data sets for the vacuum Einstein field equations, which are conformally related to static initial data sets in such a way that in a neighbourhood of infinity the two initial data sets have the same massless part. It is shown that for this class of data, the solutions to the regular finite initial value problem at spatial infinity for the conformal Einstein field equations extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data sets coincide with static data in a neighbourhood of infinity. This result highlights the special role played by static data among the class of initial data sets for the Einstein field equations whose development gives rise to a spacetime with a smooth conformal compactification at null infinity. Content Type Journal Article Pages 1-35 DOI 10.1007/s00023-011-0122-3 Authors Juan Antonio Valiente Kroon, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS UK Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Random skew plane partitions of large size distributed according to an appropriately scaled Schur process develop limit shapes. In the present work, we consider the limit of large random skew plane partitions where the inner boundary approaches a piecewise linear curve with non-lattice slopes, describing the limit shape and the local fluctuations in various regions. This analysis is fairly similar to that in Okounkov and Reshetikhin (Commun Math Phys 269:571–609, 2007 ), but we do find some new behavior. For instance, the boundary of the limit shape is now a single smooth (not algebraic) curve, whereas the boundary in Okounkov and Reshetikhin (Commun Math Phys 269:571–609, 2007 ) is singular. We also observe the bead process introduced in Boutillier (Ann Probab 37(1):107–142, 2009 ) appearing in the asymptotics at the top of the limit shape. Content Type Journal Article Pages 1-26 DOI 10.1007/s00023-011-0120-5 Authors Cedric Boutillier, UPMC Université Paris 06, UMR 7599, LPMA, 75005 Paris, France Sevak Mkrtchyan, Math Department-MS136, Rice University, 6100 S. Main St., Houston, TX 77005, USA Nicolai Reshetikhin, Department of Mathematics, UC Berkeley, 970 Evans Hall 3840, Berkeley, CA 94720, USA Peter Tingley, MIT Department of Mathematics, 77 Massachusetts Ave., Cambridge, MA 02139, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We prove that the ground state energy of an atom confined to two dimensions with an infinitely heavy nucleus of charge Z  〉 0 and N quantum electrons of charge −1 is E ( N , Z )= - \frac12 Z 2 ln Z +( E TF ( l )+\frac12 c H ) Z 2 + o ( Z 2 ) when Z → ∞ and N / Z → λ, where E TF (λ) is given by a Thomas–Fermi type variational problem and c H ≈ −2.2339 is an explicit constant. We also show that the radius of a two-dimensional neutral atom is unbounded when Z → ∞, which is contrary to the expected behavior of three-dimensional atoms. Content Type Journal Article Pages 1-30 DOI 10.1007/s00023-011-0123-2 Authors Phan Thanh Nam, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark Fabian Portmann, Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25, 10044 Stockholm, Sweden Jan Philip Solovej, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider co-rotational wave maps from (3 + 1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f 0 is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f 0 . This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in the companion paper (Donninger in Commun. Pure Appl. Math., 64(8), 2011 ). In particular, we prove that f 0 is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than \frac12 . The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes. Content Type Journal Article Pages 1-42 DOI 10.1007/s00023-011-0125-0 Authors Roland Donninger, Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60637, USA Birgit Schörkhuber, Faculty of Mathematics and Geoinformation, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8, 1040 Wien, Austria Peter C. Aichelburg, Fakultät für Physik, Gravitational Physics, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in L 2 by C Ö   e   , e being the Planck constant. Finally we present a full formal asymptotic expansion. Content Type Journal Article Pages 1-22 DOI 10.1007/s00023-011-0115-2 Authors Agissilaos Athanassoulis, DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA UK Thierry Paul, CNRS and CMLS, École Polytechnique, 91128 Palaiseau Cedex, France Federica Pezzotti, Departamento de Matemáticas, Universidad del País Vasco, Bilbao, Spain Mario Pulvirenti, Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Rome, Italy Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    For locally constant cocycles defined on an aperiodic subshift, Damanik and Lenz (Duke Math J 133(1): 95–123, 2006 ) proved that if the subshift satisfies a certain condition ( B ), then the cocycle is uniform. In this paper, we study simple Toeplitz subshifts. We give a criterion that simple Toeplitz subshifts satisfy condition ( B ), and also give some sufficient conditions that they do not satisfy condition ( B ). However, we can still prove the uniformity of Schrödinger cocycles over any simple Toeplitz subshift. As a consequence, the related Schrödinger operators have Cantor spectrum of Lebesgue measure 0. We also exhibit a fine structure for the spectrum, and this helps us to prove purely singular continuous spectrum for a large class of simple Toeplitz potentials. Content Type Journal Article Pages 1-20 DOI 10.1007/s00023-010-0075-y Authors Qing-Hui Liu, Department of Computer Science and Engineering, Beijing Institute of Technology, Beijing, 100081 People’s Republic of China Yan-Hui Qu, Department of Mathematics, Tsinghua University, Beijing, 100084 People’s Republic of China Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We investigate spectral properties of an effective Hamiltonian which is obtained as a scaling limit of the Pauli–Fierz model in nonrelativistic quantum electrodynamics. The Lamb shift of a hydrogen-like atom is derived as the lowest order approximation (in the fine structure constant) of an energy level shift of the effective Hamiltonian. Content Type Journal Article Pages 1-34 DOI 10.1007/s00023-010-0071-2 Authors Asao Arai, Department of Mathematics, Hokkaido University, Sapporo, 060-0810 Japan Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In axial symmetry, there is a gauge for Einstein equations such that the total mass of the spacetime can be written as a conserved, positive definite, integral on the spacelike slices. This property is expected to play an important role in the global evolution. In this gauge the equations reduce to a coupled hyperbolic–elliptic system which is formally singular at the axis. Due to the rather peculiar properties of the system, the local in time existence has proved to resist analysis by standard methods. To analyze the principal part of the equations, which may represent the main source of the difficulties, we study linear perturbation around the flat Minkowski solution in this gauge. In this article we solve this linearized system explicitly in terms of integral transformations in a remarkable simple form. This representation is well suited to obtain useful estimates to apply in the non-linear case. Content Type Journal Article Pages 1-17 DOI 10.1007/s00023-010-0074-z Authors Sergio Dain, Facultad de Matemática, Astronomía y Física, FaMAF, Universidad Nacional de Córdoba, Instituto de Física Enrique Gaviola, IFEG, CONICET, Ciudad Universitaria, 5000 Córdoba, Argentina Martín Reiris, Max Planck Institute for Gravitational Physics, (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We give pole free strips and estimates for resolvents of semiclassical operators which, on the level of the classical flow, have normally hyperbolic smooth trapped sets of codimension two in phase space. Such trapped sets are structurally stable and our motivation comes partly from considering the wave equation for Kerr black holes and their perturbations, whose trapped sets have precisely this structure. We give applications including local smoothing effects with epsilon derivative loss for the Schrödinger propagator as well as local energy decay results for the wave equation. Content Type Journal Article Pages 1-37 DOI 10.1007/s00023-011-0108-1 Authors Jared Wunsch, Department of Mathematics, Northwestern University, 2033, Sheridan Road, Evanston, IL 60208-2730, USA Maciej Zworski, Mathematics Department, University of California Berkeley, 2135, California Street, Berkeley, CA 94720, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We provide an explicit combinatorial expansion for the ground state energy of the massless spin-Boson model as a power series in the coupling parameter. Our method uses the technique of cluster expansion in constructive quantum field theory and takes as a starting point the functional integral representation and its reduction to an Ising model on the real line with long range interactions. We prove the analyticity of our expansion and provide an explicit lower bound on the radius of convergence. We do not need multiscale nor renormalization group analysis. A connection to the loop-erased random walk is indicated. Content Type Journal Article Pages 1-27 DOI 10.1007/s00023-011-0103-6 Authors Abdelmalek Abdesselam, Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, VA 22904-4137, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We prove that the 7oa class (equational variety) of generalized orthoarguesian lattices is properly included in all n oa classes for n  〈 7. This result strengthens the conjecture that any generalized orthoarguesian equation is strictly stronger than those of lower orders. The result emerged from our recent analysis of whether three-dimensional Kochen–Specker sets can be represented by Greechie lattices, which are a kind of orthomodular lattice. Content Type Journal Article Pages 1-13 DOI 10.1007/s00023-011-0109-0 Authors Norman D. Megill, Boston Information Group, 19 Locke Ln., Lexington, MA 02420, USA Mladen Pavičić, Institute for Theoretical Atomic, Molecular, and Optical Physics, Physics Department at Harvard University and Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We construct and discuss Hadamard states for both scalar and Dirac spinor fields in a large class of spatially flat Friedmann–Robertson–Walker spacetimes characterised by an initial phase either of exponential or of power-law expansion. The states we obtain can be interpreted as being in thermal equilibrium at the time when the scale factor a has a specific value a = a 0 . In the case a 0 = 0, these states fulfil a strict KMS condition on the boundary of the spacetime, which is either a cosmological horizon or a Big Bang hypersurface. Furthermore, in the conformally invariant case, they are conformal KMS states on the full spacetime. However, they provide a natural notion of an approximate KMS state also in the remaining cases, especially for massive fields. On the technical side, our results are based on a bulk-to-boundary reconstruction technique already successfully applied in the scalar case and here proven to be suitable also for spinor fields. The potential applications of the states we find range over a broad spectrum, but they appear to be suited to discuss in particular thermal phenomena such as the cosmic neutrino background or the quantum state of dark matter. Content Type Journal Article Pages 1-41 DOI 10.1007/s00023-011-0111-6 Authors Claudio Dappiaggi, II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Thomas-Paul Hack, II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Nicola Pinamonti, Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider the non-relativistic Hartree model in the gravitational case, i.e. with attractive Coulomb–Newton interaction. For a given mass M  〉 0, we construct stationary states with non-zero temperature T by minimizing the corresponding free energy functional. It is proved that minimizers exist if and only if the temperature of the system is below a certain threshold T * 〉 0 (possibly infinite), which itself depends on the specific choice of the entropy functional. We also investigate whether the corresponding minimizers are mixed or pure quantum states and characterize a critical temperature T c Î (0, T *) above which mixed states appear. Content Type Journal Article Pages 1-25 DOI 10.1007/s00023-011-0096-1 Authors Gonca L. Aki, Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany Jean Dolbeault, Ceremade (UMR CNRS no. 7534), Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16, France Christof Sparber, Department of Mathematics, Statistics, and Computer Science M/C 249, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In the present paper, we study forward quantum Markov chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY -model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two distinct QMC for the given family of interaction operators { K á x , y ñ } . Content Type Journal Article Pages 1-36 DOI 10.1007/s00023-011-0107-2 Authors Luigi Accardi, Centro Interdisciplinare Vito Volterra, II Università di Roma “Tor Vergata”, Via Columbia 2, 00133 Rome, Italy Farrukh Mukhamedov, Department of Computational and Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P. O. Box 141, 25710 Kuantan, Pahang, Malaysia Mansoor Saburov, Department of Computational and Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P. O. Box 141, 25710 Kuantan, Pahang, Malaysia Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    A previously proposed algebra of asymptotic fields in quantum electrodynamics is formulated as a net of algebras localized in regions which in general have unbounded spacelike extension. Electromagnetic fields may be localized in ‘symmetrical spacelike cones’, but there are strong indications this is not possible in the present model for charged fields, which have tails extending in all space directions. Nevertheless, products of appropriately ‘dressed’ fermion fields (with compensating charges) yield bi-localized observables. Content Type Journal Article Pages 1-29 DOI 10.1007/s00023-011-0105-4 Authors Andrzej Herdegen, Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Cracow, Poland Katarzyna Rejzner, Institute for Theoretical Physics, Hamburg University, Luruper Chaussee 149, 22761 Hamburg, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Let B  = ( B 1 ( t ), . . . , B d ( t )) be a d -dimensional fractional Brownian motion with Hurst index α  ≤ 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B , or to solving differential equations driven by B . We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using “standard” tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists; it is only heuristic, in particular because the desingularization of iterated integrals is really a non-perturbative effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory and stochastic calculus. The interested reader should read for a second time the companion article (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (II) The rough path for \frac16 〈 a 〈 \frac14 : constructive proof of convergence, 2011 , preprint) for the constructive proofs. Content Type Journal Article Pages 1-28 DOI 10.1007/s00023-011-0106-3 Authors Jacques Magnen, Centre de Physique Théorique, CNRS UMR 7644, École Polytechnique, 91128 Palaiseau Cedex, France Jérémie Unterberger, Institut Élie Cartan Nancy, Université Henri Poincaré, BP 239, 54506 Vandoeuvre-lès-Nancy Cedex, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider Hermitian and symmetric random band matrices H in d \geqslant 1 dimensions. The matrix elements H xy , indexed by x , y Î L Ì \mathbb Z d , are independent and their variances satisfy s xy 2 :=\mathbb E | H xy | 2 = W - d f (( x - y )/ W ) for some probability density f . We assume that the law of each matrix element H xy is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales t 〈〈 W d /3 . We also show that the localization length of the eigenvectors of H is larger than a factor W d /6 times the band width W . All results are uniform in the size |Λ| of the matrix. This extends our recent result (Erdős and Knowles in Commun. Math. Phys., 2011 ) to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying å x   s xy 2 =1 for all y , the largest eigenvalue of H is bounded with high probability by 2 + M - 2/3 + e for any $${\varepsilon 〉 0}$$, where M : = 1 / ( max x , y   s xy 2 ) . Content Type Journal Article Pages 1-93 DOI 10.1007/s00023-011-0104-5 Authors László Erdős, Institute of Mathematics, University of Munich, Theresienstr. 39, 80333 Munich, Germany Antti Knowles, Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We propose a new proof, as well as a generalization of Mirzakhani’s recursion for volumes of moduli spaces. We interpret those recursion relations in terms of expectation values in Kontsevich’s integral, i.e., we relate them to a ribbon graph decomposition of Riemann surfaces. We find a generalization of Mirzakhani’s recursions to measures containing all higher Mumford’s κ classes, and not only κ 1 as in the Weil–Petersson case. Content Type Journal Article Pages 1-17 DOI 10.1007/s00023-011-0113-4 Authors Bertrand Eynard, Service de Physique Théorique de Saclay, 91191 Gif-sur-Yvette Cedex, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider smooth (not necessarily invertible) maps of Hilbert spaces preserving ergodic Borel probability measures, and prove the existence of hyperbolic periodic orbits and horseshoes in the absence of zero Lyapunov exponents. These results extend Katok’s work on diffeomorphisms of compact manifolds to infinite dimensions, with potential applications to some classes of periodically forced PDEs. Content Type Journal Article Pages 1-28 DOI 10.1007/s00023-011-0100-9 Authors Zeng Lian, Courant Institute of Mathematical Sciences, New York University, New York, USA Lai-Sang Young, Courant Institute of Mathematical Sciences, New York University, New York, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We study the associativity property of field algebras. After extending the notion of associativity to fields of several variables and developing some structure theorems, we define meromorphic field algebras and relate them to formally rational deformation operads. Content Type Journal Article Pages 1-24 DOI 10.1007/s00023-011-0092-5 Authors Namhoon Kim, Department of Mathematics Education, Hongik University, 72-1 Sangsu-dong, Mapo-gu, Seoul, Korea Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We investigate the edge conductance of particles submitted to an Iwatsuka magnetic field, playing the role of a purely magnetic barrier. We also consider magnetic guides generated by generalized Iwatsuka potentials. In both cases, we prove quantization of the edge conductance. Next, we consider magnetic perturbations of such magnetic barriers or guides and prove stability of the quantized value of the edge conductance. Further, we establish a sum rule for edge conductances. Regularization within the context of disordered systems is discussed as well. Content Type Journal Article Pages 1-29 DOI 10.1007/s00023-011-0093-4 Authors Nicolas Dombrowski, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago de Chile, Chile François Germinet, Département de Mathématiques, Université de Cergy-Pontoise, CNRS UMR 8088, IUF, 95000 Cergy-Pontoise, France Georgi Raikov, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago de Chile, Chile Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We define a new topological polynomial extending the Bollobás–Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse–Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided. Content Type Journal Article Pages 1-63 DOI 10.1007/s00023-011-0087-2 Authors Thomas Krajewski, Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, 91405 Orsay Cedex, France Vincent Rivasseau, Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, 91405 Orsay Cedex, France Fabien Vignes-Tourneret, Institut Camille Jordan, CNRS UMR 5208, Université Claude Bernard Lyon 1, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We present a rigorous mathematical treatment of the zero-field orbital magnetic susceptibility of a non-interacting Bloch electron gas, at fixed temperature and density, for both metals and semiconductors/insulators. In particular, we obtain the Landau-Peierls formula in the low temperature and density limit as conjectured by Kjeldaas and Kohn (Phys Rev 105:806–813, 1957 ). Content Type Journal Article Pages 1-40 DOI 10.1007/s00023-011-0128-x Authors Philippe Briet, Université du sud Toulon-Var & Centre de Physique Théorique, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France Horia D. Cornean, Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg, Denmark Baptiste Savoie, Université du sud Toulon-Var & Centre de Physique Théorique, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We conclude our analysis of bubble divergences in the flat spinfoam model. In Bonzom and Smerlak, Comm. Math. Phys., ( submitted ), we showed that the divergence degree of an arbitrary 2-complex Γ can be evaluated exactly by means of twisted cohomology. Here, we specialize this result to the case where Γ is the 2-skeleton of the cell decomposition of a pseudomanifold, and sharpen it with a careful analysis of the cellular and topological structures involved. Moreover, we explain in detail how this approach reproduces all the previous powercounting results for the Boulatov–Ooguri (colored) tensor models, and sheds light on algebraic-topological aspects of Gurau’s 1/ N expansion. Content Type Journal Article Pages 1-24 DOI 10.1007/s00023-011-0127-y Authors Valentin Bonzom, Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, ON N2L 2Y5, Canada Matteo Smerlak, Centre de Physique Théorique, Campus de Luminy, Case 907, 13288 Marseille Cedex 09, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We establish several comparison results on the eigenvalue gap for Schrödinger operators on the real line. The potentials we consider here include symmetric single-well, double-well, U c -class, M c -class as well as their perturbations. Some related results on the eigenvalue ratio are also discussed. Content Type Journal Article Pages 1-17 DOI 10.1007/s00023-011-0126-z Authors Duo-Yuan Chen, Department of Mathematics, National Tsing Hua University, Hsinchu, 30043 Taiwan Min-Jei Huang, Department of Mathematics, National Tsing Hua University, Hsinchu, 30043 Taiwan Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In this second paper, we study the case of substitution tilings of \mathbb R d . The substitution on tiles induces substitutions on the faces of the tiles of all dimensions j  = 0, . . . , d − 1. We reconstruct the tiling’s equivalence relation in a purely combinatorial way using the AF-relations given by the lower dimensional substitutions. We define a Bratteli multi-diagram B which is made of the Bratteli diagrams B j , j =0, ¼ d , of all those substitutions. The set of infinite paths in B d is identified with the canonical transversal Ξ of the tiling. Any such path has a “border”, which is a set of tails in B j for some j ≤  d , and this corresponds to a natural notion of border for its associated tiling. We define an étale equivalence relation R B on B by saying that two infinite paths are equivalent if they have borders which are tail equivalent in B j for some j ≤ d . We show that R B is homeomorphic to the tiling’s equivalence relation R X . Content Type Journal Article Pages 1-36 DOI 10.1007/s00023-011-0121-4 Authors Antoine Julien, Department of Mathematics and Statistics, University of Victoria, PO Box 3060, STN CSC, Victoria, BC V8W 3R4, Canada Jean Savinien, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n -th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph’s eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem. Content Type Journal Article Pages 1-40 DOI 10.1007/s00023-011-0124-1 Authors Ram Band, Department of Mathematics, University of Bristol, Bristol, BS8 1TW UK Gregory Berkolaiko, Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA Uzy Smilansky, Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot, 76100 Israel Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We do the spectral analysis of the Hamiltonian for the weak leptonic decay of the gauge bosons W ± . Using Mourre theory, it is shown that the spectrum between the unique ground state and the first threshold is purely absolutely continuous. Neither sharp neutrino high-energy cutoff nor infrared regularization is assumed. Content Type Journal Article Pages 1-32 DOI 10.1007/s00023-011-0114-3 Authors Walter H. Aschbacher, Centre de Mathématiques Appliquées, UMR 7641, École Polytechnique-CNRS, Route de Saclay, 91128 Palaiseau Cedex, France Jean-Marie Barbaroux, Centre de Physique Théorique, Luminy Case 907, 13288 Marseille Cedex 9, France Jérémy Faupin, Institut de Mathématiques de Bordeaux , UMR-CNRS 5251, Université de Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France Jean-Claude Guillot, Centre de Mathématiques Appliquées, UMR 7641, École Polytechnique-CNRS, Route de Saclay, 91128 Palaiseau Cedex, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    In this paper we generalize the results of Gurau (arXiv:1011. 2726 [gr-qc], 2011 ), Gurau and Rivasseau (arXiv:1101.4182 [gr-qc], 2011 ) and derive the full 1/ N expansion of colored tensor models in arbitrary dimensions. We detail the expansion for the independent identically distributed model and the topological Boulatov Ooguri model. Content Type Journal Article Pages 1-25 DOI 10.1007/s00023-011-0118-z Authors Razvan Gurau, Perimeter Institute for Theoretical Physics, 31 Caroline St., Waterloo, ON, Canada Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region much larger than the one provided by the Cauchy–Kowalevski theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove this result we first describe a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering, in a gauge characterized by the introduction of a double null cone foliation of the spacetime. Then we prove that the existence region for the analytic solutions can be extended to a larger region which depends only on the validity of the a priori estimates for the Weyl equations, associated with the “Bel-Robinson norms”. In particular, if the initial data are sufficiently small we show that the analytic solution is global. Before showing how to extend the existence region we describe the same result in the case of the Burger equation, which, even if much simpler, nevertheless requires analogous logical steps required for the general proof. Due to length of this work, in this paper we mainly concentrate on the definition of the gauge we use and on writing in a “geometric” way the Einstein equations, then we show how the Cauchy–Kowalevski theorem is adapted to the characteristic problem for the Einstein equations and we describe how the existence region can be extended in the case of the Burger equation. Finally, we describe the structure of the extension proof in the case of the Einstein equations. The technical parts of this last result is the content of a second paper. Content Type Journal Article Pages 1-64 DOI 10.1007/s00023-011-0151-y Authors Giulio Caciotta, Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy Francesco Nicolò, Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface H so that restrictions f j | H to H of Δ-eigenfunctions of Riemannian manifolds ( M , g ) with ergodic geodesic flow are quantum ergodic on H . We prove two kinds of results: First (i) for any smooth hypersurface H in a piecewise-analytic Euclidean domain, the Cauchy data ( f j | H , ¶ n H f j | H ) is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly, (ii) we give conditions on H so that the Dirichlet (or Neumann) data is individually quantum ergodic. The condition involves the almost nowhere equality of left and right Poincaré maps for H . The proof involves two further novel results: (iii) a local Weyl law for boundary traces of eigenfunctions, and (iv) an ‘almost-orthogonality’ result for Fourier integral operators whose canonical relations almost nowhere commute with the geodesic flow. Content Type Journal Article Pages 1-72 DOI 10.1007/s00023-011-0154-8 Authors John A. Toth, Department of Mathematics and Statistics, McGill University, Montreal, Canada Steve Zelditch, Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We deal with a family of generalized coherent states associated to the hyperbolic Landau levels of the Schrödinger operator with uniform magnetic field on the Poincaré disk. Their associated coherent state transforms constitute a class of generalized second Bargmann transforms. Content Type Journal Article Pages 1-12 DOI 10.1007/s00023-011-0131-2 Authors Fouzia El Wassouli, Department of Mathematics, Faculty of Sciences, Ibn Tofaïl University, Kenitra, Morocco Allal Ghanmi, Department of Mathematics, Faculty of Sciences, Mohammed V University, Agdal, P. O. Box 1014, 10 000 Rabat, Morocco Ahmed Intissar, Department of Mathematics, Faculty of Sciences, Mohammed V University, Agdal, P. O. Box 1014, 10 000 Rabat, Morocco Zouhaïr Mouayn, Department of Mathematics, Faculty of Sciences and Technics (M’Ghila), Sultan Moulay Slimane University, P. O. Box 523, 23 000 Béni Mellal, Morocco Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    From the Holst action in terms of complex valued Ashtekar variables, additional reality conditions mimicking the linear simplicity constraints of spin-foam gravity are found. In quantum theory with the results of Ding and Rovelli, we are able to implement these constraints weakly, that is in the sense of Gupta and Bleuler. The resulting kinematical Hilbert space matches the original one of loop quantum gravity, that is for real valued Ashtekar connection. Our results perfectly fit with recent developments of Rovelli and Speziale concerning Lorentz covariance within spin-form gravity. Content Type Journal Article Pages 1-24 DOI 10.1007/s00023-011-0134-z Authors Wolfgang M. Wieland, Centre de Physique Théorique, Campus de Luminy, Case 907, 13288 Marseille, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    I consider random Schrödinger operators with exponentially decaying single site potential, which is allowed to change sign. For this model, I prove Anderson localization both in the sense of exponentially decaying eigenfunctions and dynamical localization. Furthermore, the results imply a Wegner-type estimate strong enough to use in classical forms of multi-scale analysis. Content Type Journal Article Pages 1-56 DOI 10.1007/s00023-011-0130-3 Authors Helge Krüger, Mathematics 253-37, Caltech, Pasadena, CA 91125, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider nonlinear elliptic Dirichlet problems with a singular term, a concave (i.e., ( p − 1)-sublinear) term and a Carathéodory perturbation. We study the cases where the Carathéodory perturbation is ( p − 1)-linear and ( p − 1)-superlinear near +∞. Using variational techniques based on the critical point theory together with truncation arguments and the method of upper and lower solutions, we show that if the L ∞ -coefficient of the concave term is small enough, the problem has at least two nontrivial smooth solutions. Content Type Journal Article Pages 1-32 DOI 10.1007/s00023-011-0129-9 Authors Leszek Gasiński, Institute of Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland Nikolaos S. Papageorgiou, Department of Mathematics, National Technical University, Zografou Campus, Athens, 15780 Greece Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We prove that the homogeneous hierarchical Anderson model exhibits a Lifshits tail at the upper edge of its spectrum. The Lifshits exponent is given in terms of the spectral dimension of the homogeneous hierarchical structure. Our approach is based on Dirichlet–Neumann bracketing for the hierarchical Laplacian and a large-deviation argument. Content Type Journal Article Pages 1-17 DOI 10.1007/s00023-011-0132-1 Authors Simon Kuttruf, Mathematisches Institut, Universität München, Theresienstraße 39, 80333 Munich, Germany Peter Müller, Mathematisches Institut, Universität München, Theresienstraße 39, 80333 Munich, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    Problems related to symmetries and dimensional reduction are common in the mathematical and physical literature, and are intensively studied presently. As a rule, the symmetry group (“reducing group”) and its orbits (“external dimensions”) are compact, and this is essential in models where the volume of the orbits is related to physical quantities. However, this case is only a part of the natural problems related to dimensional reduction. In the present paper, we consider an action of a (generally non-compact) Lie group on a vector bundle, construct a formalism of reduced bundles for description of all invariant sections of the original bundle, and study the algebraic structures that occur in the reduced bundle. We show that in the case of a non-compact reducing group it is possible that the reduction is non-standard (“non-canonical”), and construct an explicit obstruction for canonical reduction in terms of cohomology of groups. We consider in detail the reduction of tangent and cotangent bundles, and show that, in general, the duality between the two is violated in the process of reduction. The reduction of the tensor product of tangent and cotangent bundles is also discussed. We construct examples of non-canonical dimensional reduction and of violation of duality between the tangent and cotangent bundles in the reduction. Content Type Journal Article Pages 1-32 DOI 10.1007/s00023-011-0133-0 Authors Petko A. Nikolov, Department of Theoretical Physics, Faculty of Physics, University of Sofia, 5 James Bourchier Blvd, 1164 Sofia, Bulgaria Nikola P. Petrov, Department of Mathematics, University of Oklahoma, 601 Elm Avenue, Norman, OK 73019, USA Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637