ALBERT

Description:    Suppose that Alice and Bob make local two-outcome measurements on a shared entangled quantum state. We show that, for all positive integers d , there exist correlations that can only be reproduced if the local Hilbert-space dimension is at least d . This establishes that the amount of entanglement required to maximally violate a Bell inequality must depend on the number of measurement settings, not just the number of measurement outcomes. We prove this result by establishing a lower bound on a new generalization of Grothendieck’s constant. Content Type Journal Article Pages 1-17 DOI 10.1007/s00220-011-1280-3 Authors Jop Briët, Centrum Wiskunde & Informatica (CWI), and University of Amsterdam, Science Park 123, 1098 SJ Amsterdam, The Netherlands Harry Buhrman, Centrum Wiskunde & Informatica (CWI), and University of Amsterdam, Science Park 123, 1098 SJ Amsterdam, The Netherlands Ben Toner, School of Physics, The University of Melbourne, Victoria, 3010 Australia Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We provide a rigorous definition of quasi-normal modes for a rotating black hole. They are given by the poles of a certain meromorphic family of operators and agree with the heuristic definition in the physics literature. If the black hole rotates slowly enough, we show that these poles form a discrete subset of \mathbb C . As an application we prove that the local energy of linear waves in that background decays exponentially once orthogonality to the zero resonance is imposed. Content Type Journal Article Pages 1-45 DOI 10.1007/s00220-011-1286-x Authors Semyon Dyatlov, Department of Mathematics, Evans Hall, University of California, Berkeley, CA 94720, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description: Erratum to: A KAM Theorem with Applications to Partial Differential Equations of Higher Dimensions Content Type Journal Article Pages 1-1 DOI 10.1007/s00220-011-1283-0 Authors Xiaoping Yuan, School of Mathematical Sciences, Fudan University, Shanghai, 200433 P.R. China Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We consider an extension of the conditional min- and max-entropies to infinite-dimensional separable Hilbert spaces. We show that these satisfy characterizing properties known from the finite-dimensional case, and retain information-theoretic operational interpretations, e.g., the min-entropy as maximum achievable quantum correlation, and the max-entropy as decoupling accuracy. We furthermore generalize the smoothed versions of these entropies and prove an infinite-dimensional quantum asymptotic equipartition property. To facilitate these generalizations we show that the min- and max-entropy can be expressed in terms of convergent sequences of finite-dimensional min- and max-entropies, which provides a convenient technique to extend proofs from the finite to the infinite-dimensional setting. Content Type Journal Article Pages 1-22 DOI 10.1007/s00220-011-1282-1 Authors Fabian Furrer, Institute for Theoretical Physics, Leibniz Universität Hannover, 30167 Hannover, Germany Johan Åberg, Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland Renato Renner, Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition of admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type inequality. At the end of this paper, we also obtain a Miyaoka-Yau type inequality in Sasakian geometry. Content Type Journal Article Pages 1-32 DOI 10.1007/s00220-011-1292-z Authors Xi Zhang, Department of Mathematics, Zhejiang University, Hangzhou, 310027 P.R. China Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We perform rigorously the charge renormalization of the so-called reduced Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac operator, describes atoms and molecules while taking into account vacuum polarization effects. We consider the total physical density ρ ph including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no ‘real’ electron. We show that ρ ph admits an asymptotic expansion to any order in powers of the physical coupling constant α ph , provided that the ultraviolet cut-off behaves as L ~ e 3 p (1 - Z 3 )/2 a ph 〉〉 1 . The renormalization parameter 0 〈  Z 3  〈 1 is defined by Z 3  =  α ph / α , where α is the bare coupling constant. The coefficients of the expansion of ρ ph are independent of Z 3 , as expected. The first order term gives rise to the well-known Uehling potential, whereas the higher order terms satisfy an explicit recursion relation. Content Type Journal Article Pages 1-33 DOI 10.1007/s00220-011-1271-4 Authors Philippe Gravejat, Centre de Mathématiques Laurent Schwartz (UMR 7640), École Polytechnique, 91128 Palaiseau Cedex, France Mathieu Lewin, CNRS & Laboratoire de Mathématiques (UMR 8088), Université de Cergy-Pontoise, 95000 Cergy-Pontoise, France Éric Séré, Ceremade (UMR 7534), Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global properties of high-density ground state configurations in bounded domains and in infinite space. Our main result is a theorem stating that for interactions having a strictly positive Fourier transform the distribution of particles tends to be uniform as the density increases, while high-density ground states show some pattern if the Fourier transform is partially negative. The latter confirms the conclusion of earlier studies by Vlasov (in J. Phys. (USSR) IX:25–40, 1945 ), Kirzhnits and Nepomnyashchii (in Sov. Phys. JETP 32:1191–1197, 1971 ), and Likos et al . (in J. Chem. Phys. 126:224502, 2007 ). Other results include the proof that there is no Bravais lattice among high-density ground states of interactions whose Fourier transform has a negative part and the potential diverges or has a cusp at zero. We also show that in the ground state configurations of the penetrable sphere model particles are superimposed on the sites of a close-packed lattice. Content Type Journal Article Pages 1-54 DOI 10.1007/s00220-011-1276-z Authors András Sütő, Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, P. O. B. 49, 1525 Budapest, Hungary Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The chiral equivariant cohomology contains and generalizes the classical equivariant cohomology of a manifold M with an action of a compact Lie group G . For any simple G , there exist compact manifolds with the same classical equivariant cohomology, which can be distinguished by this invariant. When M is a point, this cohomology is an interesting conformal vertex algebra whose structure is still mysterious. In this paper, we scratch the surface of this object in the case G  =  SU (2). Content Type Journal Article Pages 1-37 DOI 10.1007/s00220-011-1284-z Authors Andrew R. Linshaw, Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, Germany Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    This paper is devoted to the study of strong or weak solutions of the Navier–Stokes equations in the case of an homogeneous initial data. The case of small initial data is discussed. For large initial data, an approximation is developed, in the spirit of a paper of Vishik and Fursikov. Qualitative convergence is obtained by use of the theory of Muckenhoupt weights. Content Type Journal Article Pages 1-24 DOI 10.1007/s00220-011-1291-0 Authors Pierre Gilles Lemarié–Rieusset, Équipe Analyse et Probabilités, EA2172, Université d’Evry Val d’Essonne, 91025 Evry cedex, France Frédéric Lelièvre, Équipe Analyse et Probabilités, EA2172, Université d’Evry Val d’Essonne, 91025 Evry cedex, France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We give a description of ergodic components of SRB measures in terms of ergodic homoclinic classes associated to hyperbolic periodic points. For transitive surface diffeomorphisms, we prove that there exists at most one SRB measure. Content Type Journal Article Pages 1-15 DOI 10.1007/s00220-011-1275-0 Authors F. Rodriguez Hertz, IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay M. A. Rodriguez Hertz, IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay A. Tahzibi, Departamento de Matemática, ICMC-USP São Carlos, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil R. Ures, IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We study non-associative twisted group algebras over (\mathbb Z 2 ) n with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: ( a 1 2 + ¼ + a N 2 ) ( b 1 2 + ¼ + b r ( N ) 2 ) = c 1 2 + ¼ + c N 2 , where c k are bilinear functions of the a i and b j and where ρ ( N ) is the Hurwitz-Radon function. The second application is the relation to Moufang loops and, in particular, to the code loops. To illustrate this relation, we provide an explicit coordinate formula for the factor set of the Parker loop. Content Type Journal Article Pages 1-36 DOI 10.1007/s00220-011-1279-9 Authors Sophie Morier-Genoud, Institut Mathématiques de Jussieu, UMR 7586, Université Pierre et Marie Curie Paris VI, 4 Place Jussieu, Case 247, 75252 Paris Cedex 05, France Valentin Ovsienko, CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    For n ≥ 7, we give the optimal estimate for the second eigenvalue of Paneitz operators for compact n -dimensional submanifolds in an ( n  +  p )-dimensional space form in terms of the mean curvature and the Q-curvature. Content Type Journal Article Pages 1-8 DOI 10.1007/s00220-011-1281-2 Authors Daguang Chen, Department of Mathematical Sciences, Tsinghua University, Beijing, 100084 P.R. China Haizhong Li, Department of Mathematical Sciences, Tsinghua University, Beijing, 100084 P.R. China Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    This paper concerns self-similar tilings of the Euclidean plane. We consider the number of occurrences of a given tile in any domain bounded by a Jordan curve. For a large class of self-similar tilings, including many well-known examples, we give estimates of the oscillation of this number of occurrences around its average frequency times the total number of tiles in the domain, which depend only on the Jordan curve. Content Type Journal Article Pages 1-16 DOI 10.1007/s00220-011-1274-1 Authors José Aliste-Prieto, Centro de Modelamiento Matemático, Universidad de Chile, Blanco Encalada 2120 7to. piso, Santiago, Chile Daniel Coronel, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, Chile Jean-Marc Gambaudo, Laboratoire J.-A. Dieudonné, Université de Nice - Sophia Antipolis, CNRS, 06108 Nice Cedex 02, France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The paper studies scaling limits of random skew plane partitions confined to a box when the inner shapes converge uniformly to a piecewise linear function V of arbitrary slopes in [−1, 1]. It is shown that the correlation kernels in the bulk are given by the incomplete Beta kernel, as expected. As a consequence it is established that the local correlation functions in the scaling limit do not depend on the particular sequence of discrete inner shapes that converge to V . A detailed analysis of the correlation kernels at the top of the limit shape, and of the frozen boundary is given. It is shown that depending on the slope of the linear section of the back wall, the system exhibits behavior observed in either Okounkov and Reshetikhin (Commun Math Phys 269(3):571–609, 2007 ) or Boutillier et al. ( http://arXiv.org/abs/0912.3968v2 [math-ph], 2009 ). Content Type Journal Article Pages 1-29 DOI 10.1007/s00220-011-1277-y Authors Sevak Mkrtchyan, Department of Mathematics, Rice University, Houston, TX 77005, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We study second order perturbation theory for embedded eigenvalues of an abstract class of self-adjoint operators. Using an extension of the Mourre theory, under assumptions on the regularity of bound states with respect to a conjugate operator, we prove upper semicontinuity of the point spectrum and establish the Fermi Golden Rule criterion. Our results apply to massless Pauli-Fierz Hamiltonians for arbitrary coupling. Content Type Journal Article Pages 1-36 DOI 10.1007/s00220-011-1278-x Authors J. Faupin, Institut de Mathématiques de Bordeaux, UMR-CNRS 5251, Université de Bordeaux 1, 351 Cours de la Libération, 33405 Talence Cedex, France J. S. Møller, Institut for Matematiske Fag, Aarhus Universitet, Ny Munkegade, 8000 Aarhus C, Denmark E. Skibsted, Institut for Matematiske Fag, Aarhus Universitet, Ny Munkegade, 8000 Aarhus C, Denmark Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Knuth’s parking scheme is a model in computer science for hashing with linear probing. One may imagine a circular parking lot with n sites; cars arrive at each site with unit rate. When a car arrives at a vacant site, it parks there; otherwise it turns clockwise and parks at the first vacant site which is found. We incorporate fires into this model by throwing Molotov cocktails on each site at a smaller rate n − α , where 0 〈  α  〈 1 is a fixed parameter. When a car is hit by a Molotov cocktail, it burns and the fire propagates to the entire occupied interval which turns vacant. We show that with high probability when n → ∞, the parking lot becomes saturated at a time close to 1 (i.e. as in the absence of fire) for α  〉 2/3, whereas for α  〈 2/3, the average occupation approaches 1 at time 1 but then quickly drops to 0 before the parking lot is ever saturated. Our study relies on asymptotics for the occupation of the parking lot without fires in certain regimes which may be of independent interest. Content Type Journal Article Pages 1-30 DOI 10.1007/s00220-011-1288-8 Authors Jean Bertoin, Laboratoire de Probabilités et Modèles Aléatoires, UPMC, Case courrier 188, 4 Place Jussieu, 75252 Paris, France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We investigate the low energy excitation spectrum of a Bose gas with weak, long range repulsive interactions. In particular, we prove that the Bogoliubov spectrum of elementary excitations with linear dispersion relation for small momentum becomes exact in the mean-field limit. Content Type Journal Article Pages 1-14 DOI 10.1007/s00220-011-1261-6 Authors Robert Seiringer, Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec H3A 2K6, Canada Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We consider the perturbed mKdV equation ¶ t u = - ¶ x ( ¶ x 2 u + 2 u 3 - b ( x , t ) u ) , where the potential b ( x , t )= b 0 ( hx , ht ), 0 〈 h 〈〈 1 , is slowly varying with a double soliton initial data. On a dynamically interesting time scale the solution is O ( h 2 ) close in H 2 to a double soliton whose position and scale parameters follow an effective dynamics, a simple system of ordinary differential equations. These equations are formally obtained as Hamilton’s equations for the restriction of the mKdV Hamiltonian to the submanifold of solitons. The interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to soliton stability is central in the proof. Content Type Journal Article Pages 1-63 DOI 10.1007/s00220-011-1252-7 Authors Justin Holmer, Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI 02912, USA Galina Perelman, Ecole Polytechnique, CMLS, 91128 Palaiseou, France Maciej Zworski, Department of Mathematics, University of California, Berkeley, CA 94720, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The partition function of N =6 supersymmetric Chern–Simons-matter theory (known as ABJM theory) on \mathbb S 3 , as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super–matrix model is closely related to a matrix model describing topological Chern–Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on AdS 4 ×\mathbb C \mathbb P 3 and gives the correct N 3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in \mathbb C \mathbb P 3 . We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi–Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two ’t Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi–Yau, and leads to an expansion around topological Chern–Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus. Content Type Journal Article Pages 1-53 DOI 10.1007/s00220-011-1253-6 Authors Nadav Drukker, Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany Marcos Mariño, Département de Physique Théorique, Université de Genève, Genève, CH-1211 Switzerland Pavel Putrov, Section de Mathématiques, Université de Genève, Genève, CH-1211 Switzerland Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We prove unique continuation properties for solutions of the evolution Schrödinger equation with time dependent potentials. As an application of our method we also obtain results concerning the possible concentration profiles of blow up solutions and the possible profiles of the traveling waves solutions of semi-linear Schrödinger equations. Content Type Journal Article Pages 1-26 DOI 10.1007/s00220-011-1256-3 Authors L. Escauriaza, Dpto. de Matemáticas, UPV/EHU, Apto. 644, 48080 Bilbao, Spain C. E. Kenig, Department of Mathematics, University of Chicago, Chicago, IL 60637, USA G. Ponce, Department of Mathematics, University of California, Santa Barbara, CA 93106, USA L. Vega, Dpto. de Matemáticas, UPV/EHU, Apto. 644, 48080 Bilbao, Spain Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In this article we consider the physical justification of the Vortex-Wave equation introduced by Marchioro and Pulvirenti (Mechanics, analysis and geometry: 200 years after Lagrange, North-Holland Delta Ser., Amsterdam, North-Holland, pp. 79–95, 1991 ), in the case of a single point vortex moving in an ambient vorticity. We consider a sequence of solutions for the Euler equation in the plane corresponding to initial data consisting of an ambient vorticity in L 1 ∩ L ∞ and a sequence of concentrated blobs which approach the Dirac distribution. We introduce a notion of a weak solution of the Vortex-Wave equation in terms of velocity (or primitive variables) and then show, for a subsequence of the blobs, the solutions of the Euler equation converge in velocity to a weak solution of the Vortex-Wave equation. Content Type Journal Article Pages 1-21 DOI 10.1007/s00220-011-1215-z Authors Clayton Bjorland, Department of Mathematics, University of Texas, Austin, TX 78712-0257, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The recent construction and analysis of deformations of quantum field theories by warped convolutions is extended to a class of curved spacetimes. These spacetimes carry a family of wedge-like regions which share the essential causal properties of the Poincaré transforms of the Rindler wedge in Minkowski space. In the setting of deformed quantum field theories, they play the role of typical localization regions of quantum fields and observables. As a concrete example of such a procedure, the deformation of the free Dirac field is studied. Content Type Journal Article Pages 1-32 DOI 10.1007/s00220-011-1210-4 Authors Claudio Dappiaggi, II. Institut für Theoretische Physik, D-22763 Hamburg, Deutschland Gandalf Lechner, Faculty of Physics, University of Vienna, A-1090 Vienna, Austria Eric Morfa-Morales, Erwin Schrödinger Institute for Mathematical Physics, Vienna, A-1090 Vienna, Austria Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellians. The Maxwellians are constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations. Content Type Journal Article Pages 1-52 DOI 10.1007/s00220-011-1207-z Authors Jared Speck, Department of Pure Mathematics & Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB United Kingdom Robert M. Strain, Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    For an infinitely renormalizable quadratic map f c : z ® z 2 + c with the sequence of renormalization periods { k m } and rotation numbers { t m  =  p m / q m }, we prove that if $${\limsup k_m^{-1} \log |p_m| 〉0 }$$, then the Mandelbrot set is locally connected at c . We prove also that if limsup | t m +1 | 1/ q m 〈 1 and q m → ∞, then the Julia set of f c is not locally connected and the Mandelbrot set is locally connected at c provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor. Content Type Journal Article Pages 1-34 DOI 10.1007/s00220-011-1228-7 Authors Genadi Levin, Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, 91904 Israel Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Oseledets regularity functions quantify the deviation of the growth associated with a dynamical system along its Lyapunov bundles from the corresponding uniform exponential growth. The precise degree of regularity of these functions is unknown. We show that for every invariant Lyapunov bundle of a volume preserving Anosov flow on a closed smooth Riemannian manifold, the corresponding Oseledets regularity functions are in L p ( m ), for some p 〉 0, where m is the probability measure defined by the volume form. We prove an analogous result for essentially bounded cocycles over volume preserving Anosov flows. Content Type Journal Article Pages 1-21 DOI 10.1007/s00220-011-1255-4 Authors Slobodan N. Simić, Department of Mathematics, San José State University, San José, CA 95192-0103, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The Quantum Reverse Shannon Theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel’s entanglement assisted classical capacity. In this paper, we provide a new proof of this theorem, which has previously been proved by Bennett, Devetak, Harrow, Shor, and Winter. Our proof has a clear structure being based on two recent information-theoretic results: one-shot Quantum State Merging and the Post-Selection Technique for quantum channels. Content Type Journal Article Pages 1-37 DOI 10.1007/s00220-011-1309-7 Authors Mario Berta, Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland Matthias Christandl, Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland Renato Renner, Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We show that, in a model of modified gravity based on the spectral action functional, there is a nontrivial coupling between cosmic topology and inflation, in the sense that the shape of the possible slow-roll inflation potentials obtained in the model from the nonperturbative form of the spectral action is sensitive not only to the geometry (flat or positively curved) of the universe, but also to the different possible non-simply connected topologies. We show this by explicitly computing the nonperturbative spectral action for some candidate flat cosmic topologies given by Bieberbach manifolds and showing that the resulting inflation potential differs from that of the flat torus by a multiplicative factor, similarly to what happens in the case of the spectral action of the spherical forms in relation to the case of the 3-sphere. We then show that, while the slow-roll parameters differ between the spherical and flat manifolds but do not distinguish different topologies within each class, the power spectra detect the different scalings of the slow-roll potential and therefore distinguish between the various topologies, both in the spherical and in the flat case. Content Type Journal Article Pages 1-29 DOI 10.1007/s00220-011-1352-4 Authors Matilde Marcolli, Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Elena Pierpaoli, Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, USA Kevin Teh, Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We prove the limiting absorption principle for a dressed electron at a fixed total momentum in the standard model of non-relativistic quantum electrodynamics. Our proof is based on an application of the smooth Feshbach-Schur map in conjunction with Mourre’s theory. Content Type Journal Article Pages 1-40 DOI 10.1007/s00220-011-1339-1 Authors T. Chen, Department of Mathematics, University of Texas at Austin, Austin, TX, USA J. Faupin, Institut de Mathématiques de Bordeaux, UMR-CNRS 5251, Université de Bordeaux 1, 351 Cours de la Libération, 33405 Talence Cedex, France J. Fröhlich, Institut für Theoretische Physik, ETH Hönggerberg, 8093 Zürich, Switzerland I. M. Sigal, Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of Franz (The Theory of Quantum Levy Processes, http://arxiv.org/abs/math/0407488v1 [math.PR], 2009 ), Schürmann (White noise on bioalgebras. Volume 1544 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1993 ) and others, and then formulate and study in details (with a number of interesting examples) a definition of quantum Brownian motions on those non-commutative manifolds (a la Connes) which are quantum homogeneous spaces of their quantum isometry groups in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009 ). We prove that bi-invariant quantum Brownian motion can be ‘deformed’ in a suitable sense. Moreover, we propose a non-commutative analogue of the well-known asymptotics of the exit time of classical Brownian motion. We explicitly analyze such asymptotics for a specific example on non-commutative two-torus A q , which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that A q is a non-commutative model of the (locally one-dimensional) ‘leaf-space’ of the Kronecker foliation. Content Type Journal Article Pages 1-36 DOI 10.1007/s00220-011-1368-9 Authors Biswarup Das, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108 India Debashish Goswami, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108 India Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We use Anti-de Sitter quantum field theory to prove a new class of identities between hypergeometric functions related to the Källén-Lehmann representation of products of two Anti-de Sitter two-point functions. A rich mathematical structure emerges. We apply our results to study the decay of unstable Anti-de Sitter particles. The total amplitude is in this case finite and Anti-de Sitter invariant. Content Type Journal Article Pages 1-37 DOI 10.1007/s00220-011-1372-0 Authors Jacques Bros, Institut de Physique Théorique, CEA, Saclay, France Henri Epstein, Institut des Hautes Études Scientifiques, 91440 Bures-sur-Yvette, France Michel Gaudin, Institut de Physique Théorique, CEA, Saclay, France Ugo Moschella, Università dell’Insubria, 22100 Como, Italy Vincent Pasquier, Institut de Physique Théorique, CEA, Saclay, France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We consider a two dimensional magnetic Schrödinger operator with a spatially stationary random magnetic field. We assume that the magnetic field has a positive lower bound and that it has Fourier modes on arbitrarily short scales. We prove the Wegner estimate at arbitrary energy, i.e. we show that the averaged density of states is finite throughout the whole spectrum. We also prove Anderson localization at the bottom of the spectrum. Content Type Journal Article Pages 1-36 DOI 10.1007/s00220-011-1373-z Authors László Erdős, Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany David Hasler, Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We give a simple and uniform construction of essentially all known deformation classes of gravitational instantons with ALF, ALG or ALH asymptotics and nonzero injectivity radius. We also construct new ALH Ricci flat Kähler metrics asymptotic to the product of a real line with a flat 3-manifold. Content Type Journal Article Pages 773-794 DOI 10.1007/s00220-011-1366-y Authors Olivier Biquard, Inst. de Math. de Jussieu, case 247, UPMC Université Paris 6, UMR 7586, 4, place Jussieu, 75005 Paris, France Vincent Minerbe, Inst. de Math. de Jussieu, case 247, UPMC Université Paris 6, UMR 7586, 4, place Jussieu, 75005 Paris, France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616 Journal Volume Volume 308 Journal Issue Volume 308, Number 3

Description:    We consider the large N limit of the Nahm transform, which relates charge N monopoles to solutions to the Nahm equation involving N × N matrices. In the large N limit the former approaches a magnetic bag, and the latter approaches a solution of the Nahm equation based on the Lie algebra of area-preserving vector fields on the 2-sphere. We show that the Nahm transform simplifies drastically in this limit. Content Type Journal Article Pages 1-24 DOI 10.1007/s00220-011-1378-7 Authors Derek Harland, Department of Mathematical Sciences, Durham University, Durham, DH1 3LE UK Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The formation and propagation of singularities for the Boltzmann equation in bounded domains has been an important question in numerical studies as well as in theoretical studies. In this paper, we consider the nonlinear Boltzmann solution near Maxwellians under in-flow, diffuse, or bounce-back boundary conditions. We demonstrate that discontinuity is created at the non-convex part of the grazing boundary, and then it propagates only along the forward characteristics inside the domain before it hits on the boundary again. Content Type Journal Article Pages 641-701 DOI 10.1007/s00220-011-1355-1 Authors Chanwoo Kim, Department of Mathematics, Brown University, Providence, RI 02917, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616 Journal Volume Volume 308 Journal Issue Volume 308, Number 3

Description:    We study the massless field on D n = D Ç \tfrac1 n Z 2 , where D Í R 2 is a bounded domain with smooth boundary, with Hamiltonian H ( h ) = å x ~ y   V ( h ( x ) - h ( y )) . The interaction V is assumed to be symmetric and uniformly convex. This is a general model for a (2 + 1)-dimensional effective interface where h represents the height. We take our boundary conditions to be a continuous perturbation of a macroscopic tilt: h ( x ) = n x · u + f ( x ) for x Î ¶ D n ,  u Î R 2 , and f : R 2 → R continuous. We prove that the fluctuations of linear functionals of h ( x ) about the tilt converge in the limit to a Gaussian free field on D , the standard Gaussian with respect to the weighted Dirichlet inner product ( f , g ) Ñ b = ó õ D   å i   b i ¶ i f i ¶ i g i for some explicit β = β ( u ). In a subsequent article, we will employ the tools developed here to resolve a conjecture of Sheffield that the zero contour lines of h are asymptotically described by SLE (4), a conformally invariant random curve. Content Type Journal Article Pages 591-639 DOI 10.1007/s00220-011-1315-9 Authors Jason Miller, Department of Mathematics, Stanford University, Stanford, CA 94305, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616 Journal Volume Volume 308 Journal Issue Volume 308, Number 3

Description: Erratum to: Unitary Representations of Super Lie Groups and Applications to the Classification and Multiplet Structure of Super Particles Content Type Journal Article Category Erratum Pages 1-2 DOI 10.1007/s00220-011-1332-8 Authors C. Carmeli, Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146 Genova, Italy G. Cassinelli, Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146 Genova, Italy A. Toigo, Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy V. S. Varadarajan, Department of Mathematics, University of California at Los Angeles, Box 951555, Los Angeles, CA 90095-1555, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We study limit shapes for dimer models on domains of the hexagonal lattice with free boundary conditions. This is equivalent to the large deviation phenomenon for a random stepped surface over domains fixed only at part of the boundary. Content Type Journal Article Pages 1-35 DOI 10.1007/s00220-011-1356-0 Authors Philippe Di Francesco, Institut de Physique Théorique du Commissariat à l’Energie Atomique, Unité de Recherche Associée du CNRS, CEA Saclay/IPhT/Bat 774, 91191 Gif sur Yvette Cedex, France Nicolai Reshetikhin, Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720-3840, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description: Erratum to: Fluctuations of the Nodal Length of Random Spherical Harmonics Content Type Journal Article Category Erratum Pages 1-2 DOI 10.1007/s00220-011-1367-x Authors Igor Wigman, Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG UK Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Gapped ground states of quantum spin systems have been referred to in the physics literature as being ‘in the same phase’ if there exists a family of Hamiltonians H ( s ), with finite range interactions depending continuously on s Î [0,1] , such that for each s , H ( s ) has a non-vanishing gap above its ground state and with the two initial states being the ground states of H (0) and H (1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that ’belong to the same phase’ are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an s -dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings’ ‘quasi-adiabatic evolution’ technique, of which we give a proof extended to infinite-dimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasi-local structure and prove that the flow, which we call the spectral flow , connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models H ( s ), 0 ≤ s  ≤ 1. Content Type Journal Article Pages 1-37 DOI 10.1007/s00220-011-1380-0 Authors Sven Bachmann, Department of Mathematics, University of California, Davis, Davis, CA 95616, USA Spyridon Michalakis, Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125, USA Bruno Nachtergaele, Department of Mathematics, University of California, Davis, Davis, CA 95616, USA Robert Sims, Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    This is the second part of a series of papers where we develop rigorous decay estimates for breather solutions of an averaged version of the non-linear Schrödinger equation. In this part we study the diffraction managed discrete non-linear Schrödinger equation, an equation which describes coupled waveguide arrays with periodic diffraction management geometries. We show that, for vanishing average diffraction, any solution f Î l 2 (\mathbb Z ) of the non-linear and non-local diffraction management equation decays super-exponentially. More precisely, we have the bound limsup | x | ® ¥   æ è ( | x | +1)ln( | x | +1) ö ø - 1    ln | f ( x ) | £ - \frac14 for any diffraction management soliton. Content Type Journal Article Pages 1-21 DOI 10.1007/s00220-011-1354-2 Authors Dirk Hundertmark, Institute for Analysis, Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany Young-Ran Lee, Department of Mathematics, Sogang University, Shinsu-dong 1, Mapo-gu, Seoul, 121-742 South Korea Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We give a representation-theoretic proof of a conjecture from Rietsch (Adv Math 217:2401–2442, 2008 ) providing integral formulas for solutions to the quantum Toda lattice in general type. This result generalizes work of Givental for SL n / B in a uniform way to arbitrary type, and can be interpreted as a kind of mirror theorem for the full flag variety G / B . We also prove the existence of a totally positive and totally negative critical point of the ‘superpotential’ in every mirror fiber. Content Type Journal Article Pages 1-27 DOI 10.1007/s00220-011-1308-8 Authors Konstanze Rietsch, Department of Mathematics, King’s College London, Strand, London, UK Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In this paper we consider a simplified two-dimensional scalar model for the formation of mesoscopic domain patterns in martensitic shape-memory alloys at the interface between a region occupied by the parent (austenite) phase and a region occupied by the product (martensite) phase, which can occur in two variants (twins). The model, first proposed by Kohn and Müller (Philos Mag A 66(5):697–715, 1992 ), is defined by the following functional: ( E )( u ) = 2 p b | | u (0,·) | | 2 [( H )\dot] 1/2 ([0, h ]) + ó õ L 0   dx ó õ h 0   dy   æ è | u x | 2 + \frac e 2 | u yy | ö ø , where u :[0, L ]×[0, h ] ® \mathbb R is periodic in y and u y = ± 1 almost everywhere. Conti (Cont Mech Therm 17(6):469–476, 2006 ) proved that if b 〉~ e L / h 2 and ( e / L ) 1/3 〈~ h / L , then the minimal specific energy scales like ~ min {( e b / L ) 1/2 , ( e / L ) 2/3 } , as ( e / L ) ® 0 . In the regime ( e b / L ) 1/2 〈〈 ( e / L ) 2/3 , we improve Conti’s results, by computing exactly the minimal energy and by proving that minimizers are periodic one-dimensional sawtooth functions. Content Type Journal Article Pages 1-27 DOI 10.1007/s00220-011-1374-y Authors Alessandro Giuliani, Dipartimento di Matematica di Roma Tre, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy Stefan Müller, Hausdorff Center for Mathematics & Institute for Applied Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the family of new Lax-Oleinik type operators with an arbitrary u Î C ( M , \mathbb R 1 ) as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the family of new Lax-Oleinik type operators with an arbitrary u Î C ( M , \mathbb R 1 ) as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case. Content Type Journal Article Pages 1-29 DOI 10.1007/s00220-011-1375-x Authors Kaizhi Wang, School of Mathematics, Jilin University, Changchun, 130012 China Jun Yan, School of Mathematical Sciences, Fudan University, Shanghai, 200433 China Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In this paper, we deal with global weak solutions of a non-conservative viscous compressible two-phase model in one space dimension. This work extends in some sense the previous work, [Bresch et al., Arch Rat Mech Anal 196:599–629, 2010 ], which provides the global existence of weak solutions in the multi-dimensional framework with 1 〈 γ ± 〈 6 assuming non-zero surface tension. In our study, we strongly improve the results by taking advantage of the one space dimension. More precisely, we obtain global existence of weak solutions without using capillarity terms and for pressure laws with the same range of coefficients as the degenerate barotropic mono-fluid system, namely γ ± 〉 1. Then we prove that any possible vacuum state has to vanish within finite time after which densities are always away from vacuum. This allows to prove that at least one phase corresponding to the global weak solution is a locally in time and space (in a sense to be defined) strong solution after the vacuum states vanish. Our paper may be understood as a non-straightforward generalization to the two-phase flow system of a previous paper [Li et al., Commun Math Phys 281(2):401–444, 2008 ], which treated the usual compressible barotropic Navier-Stokes equations for mono-fluid with a degenerate viscosity. Various important mathematical difficulties occur when we want to generalize those results to the two-phase flows system since the corresponding model is non-conservative. Far from vacuum, it involves a strong coupling between a nonlinear algebraic system and a degenerate PDE system under constraint linked to fractions. Moreover, fractional densities may vanish if densities or fractions vanish: A difficulty is to find estimates on the densities from estimates on fractional densities using the algebraic system. Original approximate systems have also to be introduced compared to the works on the degenerate barotropic mono-fluid system. Note that even if our result concerns “only” the one-dimensional case, it points out possible global weak solutions (for such a non-conservative system) candidates to approach for instance shock structures and to define an appropriate a priori family of paths in the phase space (in numerical schemes) at the zero dissipation limit. Content Type Journal Article Pages 1-19 DOI 10.1007/s00220-011-1379-6 Authors Didier Bresch, Laboratoire de Mathématiques, Université of Savoie, Le Bourget Du Lac, 73376 France Xiangdi Huang, Laboratoire de Mathématiques, Université of Savoie, Le Bourget Du Lac, 73376 France Jing Li, Laboratoire de Mathématiques, Université of Savoie, Le Bourget Du Lac, 73376 France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Brolin-Lyubich measure λ R of a rational endomorphism R : ^ \mathbb C   ® ^ \mathbb C   with deg R  ≥ 2 is the unique invariant measure of maximal entropy h l R = h top ( R )=log d . Its support is the Julia set J ( R ). We demonstrate that λ R is always computable by an algorithm which has access to coefficients of R , even when J ( R ) is not computable. In the case when R is a polynomial, the Brolin-Lyubich measure coincides with the harmonic measure of the basin of infinity. We find a sufficient condition for computability of the harmonic measure of a domain, which holds for the basin of infinity of a polynomial mapping, and show that computability may fail for a general domain. Content Type Journal Article Pages 1-29 DOI 10.1007/s00220-011-1363-1 Authors Ilia Binder, Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada Mark Braverman, Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada Cristobal Rojas, Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada Michael Yampolsky, Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in Doplicher et al. (Commun Math Phys 172:187–220, 1995 ). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes. The Minkowski distance operator between two independent events is shown to have pure Lebesgue spectrum with infinite multiplicity. The Euclidean distance operator is shown to have spectrum bounded below by a constant of the order of the Planck length. The corresponding statement is proved also for both the space-space and space-time area operators, as well as for the Euclidean length of the vector representing the 3-volume operators. However, the space 3-volume operator (the time component of that vector) is shown to have spectrum equal to the whole complex plane. All these operators are normal, while the distance operators are also selfadjoint. The Lorentz invariant spacetime volume operator, representing the 4-volume spanned by five independent events, is shown to be normal. Its spectrum is pure point with a finite distance (of the order of the fourth power of the Planck length) away from the origin. The mathematical formalism apt to these problems is developed and its relation to a general formulation of Gauge Theories on Quantum Spaces is outlined. As a byproduct, a Hodge Duality between the absolute differential and the Hochschild boundary is pointed out. Content Type Journal Article Pages 1-23 DOI 10.1007/s00220-011-1358-y Authors D. Bahns, Courant Research Center Mathematics, University of Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany S. Doplicher, Dipartimento di Matematica, Università degli Studi di Roma “Sapienza”, P.le A. Moro 2, 00185 Roma, Italy K. Fredenhagen, II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany G. Piacitelli, SISSA – Via Bonomea 265, 34136 Trieste, Italy Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We study one–dimensional Ising spin systems with ferromagnetic, long–range interaction decaying as n −2+ α , a Î [0,\frac 12] , in the presence of external random fields. We assume that the random fields are given by a collection of symmetric, independent, identically distributed real random variables, which are gaussian or subgaussian with variance θ . We show that when the temperature and the variance of the randomness are sufficiently small, with overwhelming probability with respect to the random fields, the typical configurations, within intervals centered at the origin whose length grow faster than any power of θ −1 , are intervals of + spins followed by intervals of − spins whose typical length is @   q - \frac2(1 - 2 a ) for 0 ≤ α  〈 1/2 and between e \frac1 q and e \frac 1 q 2 for α  = 1/2. Content Type Journal Article Pages 1-25 DOI 10.1007/s00220-011-1371-1 Authors Marzio Cassandro, Dipartimento di Fisica, Universitá di Roma “La Sapienza”, P.le A. Moro, 00185 Roma, Italy Enza Orlandi, Dipartimento di Matematica, Universitá di Roma Tre, L.go S.Murialdo 1, 00146 Roma, Italy Pierre Picco, LATP, CMI, UMR 6632, CNRS, Université de Provence, 39 rue Frederic Joliot Curie, 13453 Marseille Cedex 13, France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We consider a large lattice system of unbounded continuous spins that are governed by a Ginzburg-Landau type potential and a weak quadratic interaction. We derive the logarithmic Sobolev inequality (LSI) for Kawasaki dynamics uniform in the boundary data. The scaling of the LSI constant is optimal in the system size and our argument is independent of the geometric structure of the system. The proof consists of an application of the two-scale approach of Grunewald, Otto, Westdickenberg & Villani. Several ideas are needed to solve new technical difficulties due to the interaction. Let us mention the application of a new covariance estimate, a conditioning technique, and a generalization of the local Cramér theorem. Content Type Journal Article Pages 1-44 DOI 10.1007/s00220-011-1326-6 Authors Georg Menz, Max Planck Institute for Mathematics in the Sciences, Inselstr. 22 - 26, Leipzig, Germany Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We show that for all q in the interval (−1, 1), the Fock representation of the q -commutation relations can be unitarily embedded into the Fock representation of the extended Cuntz algebra. In particular, this implies that the C *-algebra generated by the Fock representation of the q -commutation relations is exact. An immediate consequence is that the q -Gaussian von Neumann algebra is weakly exact for all q in the interval (−1, 1). Content Type Journal Article Pages 1-18 DOI 10.1007/s00220-011-1323-9 Authors Matthew Kennedy, Pure Mathematics Department, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Alexandru Nica, Pure Mathematics Department, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We define and study on Lorentz manifolds a family of covariant diffusions in which the quadratic variation is locally determined by the curvature. This allows the interpretation of the diffusion effect on a particle by its interaction with the ambient space-time. We will focus on the case of warped products, especially Robertson-Walker manifolds, and analyse their asymptotic behaviour in the case of Einstein-de Sitter-like manifolds. Content Type Journal Article Pages 1-32 DOI 10.1007/s00220-011-1312-z Authors Jacques Franchi, IRMA, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France Yves Le Jan, Université Paris Sud 11, Mathématiques, Bâtiment 425, 91405 Orsay, France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We consider the nonlinear Schrödinger equation ( - D + V ( x )) u = G ( x ) | u | p - 1 u ,     x Î \mathbb R n with V ( x ) = V 1 ( x ), Γ( x ) = Γ 1 ( x ) for x 1  〉 0 and V ( x ) = V 2 ( x ), Γ( x ) = Γ 2 ( x ) for x 1  〈 0, where V 1 , V 2 , Γ 1 , Γ 2 are periodic in each coordinate direction. This problem describes the interface of two periodic media, e.g. photonic crystals. We study the existence of ground state H 1 solutions (surface gap soliton ground states) for 0 〈 min σ (−Δ +  V ). Using a concentration compactness argument, we provide an abstract criterion for the existence based on ground state energies of each periodic problem (with V ≡ V 1 , Γ ≡ Γ 1 and V ≡ V 2 , Γ ≡ Γ 2 ) as well as a more practical criterion based on ground states themselves. Examples of interfaces satisfying these criteria are provided. In 1D it is shown that, surprisingly, the criteria can be reduced to conditions on the linear Bloch waves of the operators - \tfrac d 2 dx 2 + V 1 ( x ) and - \tfrac d 2 dx 2 + V 2 ( x ) . Content Type Journal Article Pages 1-32 DOI 10.1007/s00220-011-1320-z Authors Tomáš Dohnal, Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany Michael Plum, Institut für Analysis, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany Wolfgang Reichel, Institut für Analysis, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data D Hg fits into a family D w Hg 2 n +1 , where n ≥  0 and w Î \mathbb Z 2 n +1 . We show D 0 Hg 2 n +1 is related to the subfactors Izumi hypothetically associates to the cyclic groups \mathbb Z 2 n +1 . Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions, etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of Izumi type \mathbb Z 7 , \mathbb Z 9 and \mathbb Z 3 2 , and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type \mathbb Z 11 ,\mathbb Z 13 ,\mathbb Z 15 ,\mathbb Z 17 ,\mathbb Z 19 (previously, Izumi had shown uniqueness for \mathbb Z 3 and \mathbb Z 5 ), and we identify their modular data. We explain how D Hg (more generally D w Hg 2 n +1 ) is a graft of the quantum double D Sym (3) (resp. the twisted double D w D 2 n +1 ) by affine so(13) (resp. so (4 n 2 +4 n +5) ) at level 2. We discuss the vertex operator algebra (or conformal field theory) realisation of the modular data D w Hg 2 n +1 . For example we show there are exactly 2 possible character vectors (giving graded dimensions of all modules) for the Haagerup VOA at central charge c = 8. It seems unlikely that any of this twisted Haagerup-Izumi modular data can be regarded as exotic, in any reasonable sense. Content Type Journal Article Pages 1-50 DOI 10.1007/s00220-011-1329-3 Authors David E. Evans, School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG Wales, UK Terry Gannon, Department of Mathematics, University of Alberta, Edmonton, AB T6G 2G1, Canada Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, which could be applied to a large class of Hamiltonian PDEs containing the derivative ∂ x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed Benjamin-Ono equation with periodic boundary conditions, so KAM tori and thus quasi-periodic solutions are obtained for them. Content Type Journal Article Pages 1-45 DOI 10.1007/s00220-011-1353-3 Authors Jianjun Liu, School of Mathematical Sciences and Key Lab of Math. for Nonlinear Science, Fudan University, Shanghai, 200433 P.R. China Xiaoping Yuan, School of Mathematical Sciences and Key Lab of Math. for Nonlinear Science, Fudan University, Shanghai, 200433 P.R. China Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In this paper, we consider the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations with initial data in the critical Besov-Sobolev type spaces B and B - \frac12,\frac12 4 (see Definitions 1.1 and 1.2 below). In particular, we proved that there exists a positive constant C such that ( ANS ν ) has a unique global solution with initial data u 0 = ( u 0 h , u 0 3 ) which satisfies || u 0 h || B exp æ è \frac C n 4 || u 0 3 || B 4 ö ø £ c 0 n or || u 0 h || B - \frac12,\frac12 4 exp æ è \frac C n 4 || u 0 3 || B - \frac12,\frac12 4 4 ö ø £ c 0 n for some c 0 sufficiently small. To overcome the difficulty that Gronwall’s inequality can not be applied in the framework of Chemin-Lerner type spaces, ~ L p t   ( B ) , we introduced here sort of weighted Chemin-Lerner type spaces, ~ L 2 t , f   ( B ) for some apropriate L 1 function f ( t ). Content Type Journal Article Pages 1-47 DOI 10.1007/s00220-011-1350-6 Authors Marius Paicu, Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 33405 Talence Cedex, France Ping Zhang, Academy of Mathematics and Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190 China Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman and Rodnianski (Hyperbolic Equ. 4(3):401–433, 2007 ) to systems of tensor wave equations with additional first-order terms. We also present a different derivation, which better highlights that such representation formulas are supported entirely on past null cones. This generalization of (Hyperbolic Equ. 4(3):401–433, 2007 ) is a key component for extending Klainerman and Rodnianski’s breakdown criterion result for Einstein-vacuum spacetimes in (J. Amer. Math. Soc. 23(2):345–382, 2009 ) to Einstein-Maxwell and Einstein-Yang-Mills spacetimes. Content Type Journal Article Pages 1-32 DOI 10.1007/s00220-011-1273-2 Authors Arick Shao, Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We consider a family of time-dependent dephasing Lindblad generators which model the monitoring of the instantaneous Hamiltonian of a system by a Markovian bath. In this family the time dependence of the dephasing operators is (essentially) governed by the instantaneous Hamiltonian. The evolution in the adiabatic limit admits a geometric interpretation and can be solved by quadrature. As an application we derive an analog of the Landau-Zener tunneling formula for this family. Content Type Journal Article Pages 1-7 DOI 10.1007/s00220-011-1269-y Authors J. E. Avron, Department of Physics, Technion, 32000 Haifa, Israel M. Fraas, Department of Physics, Technion, 32000 Haifa, Israel G. M. Graf, Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland P. Grech, Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In n -dimensional Euclidean space let us be given an infinitely differentiable real valued function V that is bounded below. We associate with the formal operator that sends a complex valued function ψ into −div(grad ψ ) +  V ψ a uniquely defined self adjoint operator which we will denote by −Δ +  V . If ψ 0 is any eigenfunction of the self adjoint operator −Δ +  V we prove that a necessary and sufficient condition for ψ 0 to never equal zero is that the eigenspace to which ψ 0 belongs contain a positive function. In this case the eigenspace must be one dimensional. The same result holds on any complete connected Riemannian manifold whose first Betti number is zero. Content Type Journal Article Pages 1-5 DOI 10.1007/s00220-011-1272-3 Authors Sol Schwartzman, Math. Department, University of Rhode Island, Kingston, RI 02881, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Recently, several authors studied small quantum systems weakly coupled to free boson or fermion fields at positive temperature. All the rigorous approaches we are aware of employ complex deformations of Liouvillians or Mourre theory (the infinitesimal version of the former). We present an approach based on polymer expansions of statistical mechanics. Despite the fact that our approach is elementary, our results are slightly sharper than those contained in the literature up to now. We show that, whenever the small quantum system is known to admit a Markov approximation (Pauli master equation aka Lindblad equation) in the weak coupling limit, and the Markov approximation is exponentially mixing, then the weakly coupled system approaches a unique invariant state that is perturbatively close to its Markov approximation. Content Type Journal Article Pages 1-30 DOI 10.1007/s00220-011-1247-4 Authors W. De Roeck, Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 19, D-69120 Heidelberg, Germany A. Kupiainen, Department of Mathematics, University of Helsinki, P. O. Box 68, Helsinki, FIN-00014 Finland Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In previous work, we introduced a class of integrable spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter, as classified by Etingof and Varchenko for simple Lie algebras. Here the main purpose is to establish the Liouville integrability of these systems by a uniform method based on evaluating the primitive invariants of Chevalley on the Lax operators with spectral parameter. As part of our analysis, we will develop several results concerning the algebra of invariant polynomials on simple Lie algebras and their expansions. Content Type Journal Article Pages 1-24 DOI 10.1007/s00220-011-1359-x Authors Luen-Chau Li, Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA Zhaohu Nie, Department of Mathematics, Pennsylvania State University, Altoona Campus, 3000 Ivyside Park, Altoona, PA 16601, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We develop a theory of f -coordinated (quasi-) modules for a general nonlocal vertex algebra where f is what we call an associate of the one-dimensional additive formal group. By specializing f to a particular associate, we obtain a new construction of weak quantum vertex algebras in the sense of Li (Selecta Mathematica (New Series) 11:349–397, 2005 ). As an application, we associate weak quantum vertex algebras to quantum affine algebras, and we also associate quantum vertex algebras and f -coordinated modules to a certain quantum βγ -system explicitly. Content Type Journal Article Pages 1-39 DOI 10.1007/s00220-011-1364-0 Authors Haisheng Li, Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We study the limit of quasilocal energy defined in Wang and Yau (Phys Rev Lett 102(2):021101, 2009 ; Commun Math Phys 288(3):919–942, 2009 ) for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi–Sachs energy-momentum at a retarded time. The quasilocal mass in Wang and Yau (Phys Rev Lett 102(2):021101, 2009 ; Commun Math Phys 288(3):919–942, 2009 ) is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases. Assuming analyticity, the equation can be solved and the solution is locally minimizing in all orders. In particular, this produces an optimal reference hypersurface in the Minkowski space for the spatial or null exterior region of an asymptotically flat spacetime. Content Type Journal Article Pages 1-19 DOI 10.1007/s00220-011-1362-2 Authors PoNing Chen, Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Mu-Tao Wang, Department of Mathematics, Columbia University, New York, NY 10027, USA Shing-Tung Yau, Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Tunnelling processes through black hole horizons have recently been investigated in the framework of WKB theory, discovering an interesting interplay with Hawking radiation. In this paper, we instead adopt the point of view proper of QFT in curved spacetime, namely, we use a suitable scaling limit towards a Killing horizon to obtain the leading order of the correlation function relevant for the tunnelling. The computation is done for a certain large class of reference quantum states for scalar fields, including Hadamard states. In the limit of sharp localization either on the external side or on opposite sides of the horizon, the quantum correlation functions appear to have thermal nature. In both cases the characteristic temperature is referred to the surface gravity associated with the Killing field and thus connected with the Hawking one. Our approach is valid for every stationary charged rotating non-extremal black hole. However, since the computation is completely local, it covers the case of a Killing horizon which just temporarily exists in some finite region, too. These results provide strong support to the idea that the Hawking radiation, which is detected at future null infinity and needs some global structures to be defined, is actually related to a local phenomenon taking place even for local geometric structures (local Killing horizons), existing just for a while. Content Type Journal Article Pages 1-17 DOI 10.1007/s00220-011-1369-8 Authors Valter Moretti, Dipartimento di Matematica, Università di Trento and Istituto Nazionale di Fisica Nucleare – Gruppo Collegato di Trento, via Sommarive 14, 38050 Povo (TN), Italy Nicola Pinamonti, Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter. Content Type Journal Article Pages 1-50 DOI 10.1007/s00220-011-1361-3 Authors E. Kirr, Department of Mathematics, University of Illinois, Urbana–Champaign, Urbana, IL 61801, USA P. G. Kevrekidis, Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA D. E. Pelinovsky, Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a transitive invariant set Ω of full Hausdorff dimension. The set Ω is a topological limit of hyperbolic sets and is accumulated by elliptic islands. As an application we prove that a stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters. Content Type Journal Article Pages 1-38 DOI 10.1007/s00220-011-1365-z Authors Anton Gorodetski, Department of Mathematics, University of California, Irvine, CA 92697, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a Kähler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU q (2), and also of invariant gauge connections on these bundles. The reduction of Yang–Mills gauge theory on the product space leads to a q -deformation of the usual quiver gauge theories on M . We formulate generalized instanton equations on the quantum space and show that they correspond to q -deformations of the usual holomorphic quiver chain vortex equations on M . We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q -deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q -deformations of instantons whose moduli spaces admit the standard hyper-Kähler quotient construction. Content Type Journal Article Pages 1-49 DOI 10.1007/s00220-011-1357-z Authors Giovanni Landi, Dipartimento di Matematica e Informatica, Università di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy Richard J. Szabo, Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh, EH14 4AS UK Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The notion of a Radon transform is introduced for completely integrable billiard tables. In the case of Liouville billiard tables of dimension 3 we prove that the Radon transform is one-to-one on the space of continuous functions K on the boundary which are invariant with respect to the corresponding group of symmetries. We prove also that the frequency map associated with a class of Liouville billiard tables is non-degenerate. This allows us to obtain spectral rigidity of the corresponding Laplace-Beltrami operator with Robin boundary conditions. Content Type Journal Article Pages 1-39 DOI 10.1007/s00220-011-1223-z Authors G. Popov, Laboratoire de Mathématiques Jean Leray, CNRS: UMR6629, Université de Nantes, 2, rue de la Houssinière, BP 92208, 44072 Nantes Cedex 03, France P. Topalov, Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In this article, we construct explicitly several holomorphic vertex operator algebras of central charge 24 using Virasoro frames. The Lie algebras associated to their weight one subspaces are of the types A 1,2 A 3,4 4 , A 1,2 D 5,8 , A 1,1 3 A 7,4 , A 1,1 2 C 3,2 D 5,4 , A 2,1 2 A 5,2 2 C 2,1 , A 3,1 A 7,2 C 3,1 2 , A 3,1 C 7,2 , A 4,1 A 9,2 B 3,1 , B 4,1 C 6,1 2 and B 6,1 C 10,1 . These vertex operator algebras correspond to number 7, 10, 18, 19, 26, 33, 35, 40, 48 and 56 in Schellekens’ list Schellekens (Commun Math Phys 153:159–185, 1993 ). Content Type Journal Article Pages 1-46 DOI 10.1007/s00220-011-1212-2 Authors Ching Hung Lam, Institute of Mathematics, Academia Sinica, Taipei, 10617 Taiwan Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The program of constructing spacetime geometry from string theoretic modular forms is extended to Calabi-Yau varieties of dimensions three and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the L –functions associated to omega motives of Calabi-Yau varieties, generated by their holomorphic n −forms via Galois representations. The modular forms that emerge in this way are related to characters of the underlying rational conformal field theory. The converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication. The emerging picture suggests that the L –function can be viewed as defining a map between the geometric category of motives and the category of conformal field theories on the worldsheet. Content Type Journal Article Pages 1-30 DOI 10.1007/s00220-010-1179-4 Authors Rolf Schimmrigk, Indiana University South Bend, 1700 Mishawaka Ave., South Bend, IN 46634, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Random matrix models generalize to Group Field Theories (GFT) whose Feynman graphs are dual to higher dimensional topological spaces. The perturbative development of the usual GFT’s is rather involved combinatorially and plagued by topological singularities (which we discuss in great detail in this paper), thus very difficult to control and unsatisfactory. Both these problems simplify greatly for the “colored” GFT (CGFT) model we introduce in this paper. Not only this model is combinatorially simpler but also it is free from the worst topological singularities. We establish that the Feynman graphs of our model are combinatorial cellular complexes dual to manifolds or pseudomanifolds, and study their cellular homology. We also relate the amplitude of CGFT graphs to their fundamental group. Content Type Journal Article Pages 1-25 DOI 10.1007/s00220-011-1226-9 Authors Razvan Gurau, Perimeter Institute for Theoretical Physics Waterloo, Waterloo, ON N2L 2Y5, Canada Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In this paper we consider pseudo-bihermitian structures – pairs of complex structures compatible with a pseudo-Riemannian metric. We establish relations of these structures with generalized (pseudo-) Kähler geometry and holomorphic Poisson structures similar to that in the positive definite case. We provide a list of compact complex surfaces which could admit pseudo-bihermitian structures and give examples of such structures on some of them. We also consider a naturally defined null plane distribution on a generalized pseudo-Kähler 4-manifold and show that under a mild restriction it determines an Engel structure. Content Type Journal Article Pages 1-20 DOI 10.1007/s00220-011-1213-1 Authors Johann Davidov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Gueo Grantcharov, Department of Mathematics, Florida International University, Miami, FL 33199, USA Oleg Mushkarov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Miroslav Yotov, Department of Mathematics, Florida International University, Miami, FL 33199, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Rigorous estimates for the total – (kinetic) energy plus pressure – flux in \mathbb R 3 are obtained from the three dimensional Navier-Stokes equations. The bounds are used to establish a condition – involving Taylor length scale and the size of the domain – sufficient for existence of the inertial range and the energy cascade in decaying turbulence (zero driving force, non-increasing global energy). Several manifestations of the locality of the flux under this condition are obtained. All the scales involved are actual physical scales in \mathbb R 3 and no regularity or homogeneity/scaling assumptions are made. Content Type Journal Article Pages 1-22 DOI 10.1007/s00220-011-1219-8 Authors R. Dascaliuc, Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA Z. Grujić, Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We use a coupling technique to prove that there exists a loop-erasure of the time-reversal of a planar Brownian motion stopped on exiting a simply connected domain, and that the loop-erased curve is a radial SLE 2 curve. This result extends to Brownian motions and Brownian excursions under certain conditioning in a finitely connected plane domain, and the loop-erased curve is a continuous LERW curve. Content Type Journal Article Pages 1-12 DOI 10.1007/s00220-011-1234-9 Authors Dapeng Zhan, Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We study finite quasi-quantum groups in their quiver setting developed recently by the first author. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a classification of the finite tensor categories in which every simple object has Frobenius-Perron dimension 1 and there are finitely many indecomposable objects up to isomorphism. Some interesting information of these finite tensor categories is given by making use of the quiver representation theory. Content Type Journal Article Pages 1-18 DOI 10.1007/s00220-011-1229-6 Authors Hua-Lin Huang, School of Mathematics, Shandong University, Jinan, Shandong 250100, China Gongxiang Liu, Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, China Yu Ye, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions. Content Type Journal Article Pages 1-8 DOI 10.1007/s00220-011-1225-x Authors Tobias H. Colding, Dept. of Math., MIT, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA William P. Minicozzi, Dept. of Math., Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors | y ñ that have a fixed expectation value á y | H | y ñ = E with respect to H . Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere. Content Type Journal Article Pages 1-40 DOI 10.1007/s00220-011-1205-1 Authors Markus P. Müller, Institute of Mathematics, Technical University of Berlin, 10623 Berlin, Germany David Gross, Institute for Theoretical Physics, Leibniz University Hannover, 30167 Hannover, Germany Jens Eisert, Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on M n (\mathbb C ) for all n ≥ 3, as well as an example of a one-parameter semigroup ( T ( t )) t ≥0 of Markov maps on M 4 (\mathbb C ) such that T ( t ) fails to be factorizable for all small values of t 〉 0. As applications, we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little Grothendieck inequality. Content Type Journal Article Pages 1-40 DOI 10.1007/s00220-011-1216-y Authors Uffe Haagerup, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark Magdalena Musat, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In [A.W. Harrow and R.A. Low, Commun. Math. Phys. 291(1):257–302 ( 2009 )], it was shown that a quantum circuit composed of random 2-qubit gates converges to an approximate quantum 2-design in polynomial time. We point out and correct a flaw in one of the paper’s main arguments. Our alternative argument highlights the role played by transpositions induced by the random gates in achieving convergence. Content Type Journal Article Pages 1-13 DOI 10.1007/s00220-011-1217-x Authors Igor Tuche Diniz, Instituto de Física, Universidade Federal Fluminense, Niterói, Brazil Daniel Jonathan, Instituto de Física, Universidade Federal Fluminense, Niterói, Brazil Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We prove that the thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. We also show that at small coupling, all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly. Moreover, for a sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz. Finally, we provide explicit upper and lower bounds for the solutions to the difference equation and use them to study the spectral measures and the transport exponents. Content Type Journal Article Pages 1-57 DOI 10.1007/s00220-011-1220-2 Authors David Damanik, Department of Mathematics, Rice University, Houston, TX 77005, USA Anton Gorodetski, Department of Mathematics, University of California, Irvine, CA 92697, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the q -Ornstein-Uhlenbeck semigroup on the q -deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the q -Araki-Woods algebras. We show that hypercontractivity from L p to L 2 can occur if and only if the generator of the deformation is bounded. Content Type Journal Article Pages 1-21 DOI 10.1007/s00220-011-1221-1 Authors Hun Hee Lee, Department of Mathematics, Chungbuk National University, 410 Sungbong-Ro, Heungduk-Gu, Cheongju, 361-763 Korea Éric Ricard, Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We give a full list of known N =1 supersymmetric quantum field theories related by the Seiberg electric-magnetic duality conjectures for SU ( N ), SP (2 N ) and G 2 gauge groups. Many of the presented dualities are new, not considered earlier in the literature. For all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals. This gives a systematic extension of the related Römelsberger and Dolan-Osborn results. Equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals. About half of them were proven earlier, and another half represents new challenging conjectures. In particular, we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author. Content Type Journal Article Pages 1-78 DOI 10.1007/s00220-011-1218-9 Authors V. P. Spiridonov, Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region, 141980 Russia G. S. Vartanov, Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We consider a heavy quantum particle with an internal degree of freedom moving on the d -dimensional lattice \mathbb Z d (e.g., a heavy atom with finitely many internal states). The particle is coupled to a thermal medium (bath) consisting of free relativistic bosons (photons or Goldstone modes) through an interaction of strength λ linear in creation and annihilation operators. The mass of the quantum particle is assumed to be of order λ −2 , and we assume that the internal degree of freedom is coupled “effectively” to the thermal medium. We prove that the motion of the quantum particle is diffusive in d  ≥ 4 and for λ small enough. Content Type Journal Article Pages 1-95 DOI 10.1007/s00220-011-1222-0 Authors W. De Roeck, Institut für Theoretische Physik, Universität Heidelberg, D 69120 Heidelberg, Germany J. Fröhlich, Institute for Theoretical Physics, ETH Zürich, CH-8093 Zürich, Switzerland Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slow-roll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternionic space, the Poincaré dodecahedral space, and the flat tori. We compute the corresponding slow-roll parameters and we check that the resulting inflation model behaves in the same way as for a simply-connected spherical topology in the case of the quaternionic space and the Poincaré homology sphere, while it behaves differently in the case of the flat tori. We add an appendix with a discussion of the case of lens spaces. Content Type Journal Article Pages 1-50 DOI 10.1007/s00220-011-1211-3 Authors Matilde Marcolli, Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Elena Pierpaoli, Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, USA Kevin Teh, Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In the framework of algebraic quantum field theory, we study the category D BF \mathfrak A of stringlike localised representations of a net of observables O ® \mathfrak A ( O ) in three dimensions. It is shown that compactly localised (DHR) representations give rise to a non-trivial centre of D BF \mathfrak A with respect to the braiding. This implies that D BF \mathfrak A cannot be modular when non-trivial DHR sectors exist. Modular tensor categories, however, are important for topological quantum computing. For this reason, we discuss a method to remove this obstruction to modularity. Indeed, the obstruction can be removed by passing from the observable net \mathfrak A ( O ) to the Doplicher-Roberts field net \mathfrak F ( O ) . It is then shown that sectors of \mathfrak A can be extended to sectors of the field net that commute with the action of the corresponding symmetry group. Moreover, all such sectors are extensions of sectors of \mathfrak A . Finally, the category D BF \mathfrak F of sectors of \mathfrak F is studied by investigating the relation with the categorical crossed product of D BF \mathfrak A by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category D BF \mathfrak F . Content Type Journal Article Pages 1-36 DOI 10.1007/s00220-011-1200-6 Authors Pieter Naaijkens, Radboud University Nijmegen, Institute for Mathematics, Astrophysics and Particle Physics, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The renormalization group (RG) approach is largely responsible for the considerable success that has been achieved in developing a quantitative theory of phase transitions. Physical properties emerge from spectral properties of the linearization of the RG map at a fixed point. This article considers RG for classical Ising-type lattice systems. The linearization acts on an infinite-dimensional Banach space of interactions. At a trivial fixed point (zero interaction), the spectral properties of the RG linearization can be worked out explicitly, without any approximation. The results are for the RG maps corresponding to decimation and majority rule. They indicate spectrum of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, only residual spectrum. This may serve as a lesson in what one might expect in more general situations. Content Type Journal Article Pages 1-12 DOI 10.1007/s00220-011-1201-5 Authors Mei Yin, Department of Mathematics, University of Arizona, Tucson, Arizona 85721, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We present a new class of near-horizon geometries which solve Einstein’s vacuum equations, including a negative cosmological constant, in all even dimensions greater than four. Spatial sections of the horizon are inhomogeneous S 2 -bundles over any compact Kähler-Einstein manifold. For a given base, the solutions are parameterised by one continuous parameter (the angular momentum) and an integer which determines the topology of the horizon. In six dimensions the horizon topology is either S 2  × S 2 or \mathbb CP 2 # \mathbb CP 2   . In higher dimensions the S 2 -bundles are always non-trivial, and for a fixed base, give an infinite number of distinct horizon topologies. Furthermore, depending on the choice of base we can get examples of near-horizon geometries with a single rotational symmetry (the minimal dimension for this is eight). All of our horizon geometries are consistent with all known topology and symmetry constraints for the horizons of asymptotically flat or globally Anti de Sitter extremal black holes. Content Type Journal Article Pages 1-41 DOI 10.1007/s00220-011-1192-2 Authors Hari K. Kunduri, Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, AB T6G 2J1, Canada James Lucietti, Theoretical Physics, Blackett Laboratory, Imperial College London, London, SW7 2AZ UK Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We prove a uniqueness theorem for stationary D -dimensional Kaluza-Klein black holes with D − 2 Killing fields, generating the symmetry group \mathbb R × U (1) D - 3 . It is shown that the topology and metric of such black holes is uniquely determined by the angular momenta and certain other invariants consisting of a number of real moduli, as well as integer vectors subject to certain constraints. Content Type Journal Article Pages 1-44 DOI 10.1007/s00220-010-1176-7 Authors Stefan Hollands, School of Mathematics, Cardiff University, Cardiff, CF24 4AG UK Stoytcho Yazadjiev, Department of Theoretical Physics, Faculty of Physics, Sofia University, 5 J. Bourchier Blvd., Sofia, 1164 Bulgaria Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    For nonlinear wave equations existence proofs for breathers are very rare. In the spatially homogeneous case up to rescaling the sine-Gordon equation ¶ 2 t u = ¶ 2 x u - sin( u ) is the only nonlinear wave equation which is known to possess breather solutions. For nonlinear wave equations in periodic media no examples of breather solutions have been known so far. Using spatial dynamics, center manifold theory and bifurcation theory for periodic systems we construct for the first time such time periodic solutions of finite energy for a nonlinear wave equation s ( x ) ¶ 2 t u ( x , t ) = ¶ 2 x u ( x , t ) - q ( x ) u ( x , t )+ r ( x ) u ( x , t ) 3 , with spatially periodic coefficients s , q , and r on the real axis. Such breather solutions play an important role in theoretical scenarios where photonic crystals are used as optical storage. Content Type Journal Article Pages 1-27 DOI 10.1007/s00220-011-1191-3 Authors Carsten Blank, Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany Martina Chirilus-Bruckner, Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany Vincent Lescarret, Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany Guido Schneider, Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In this paper we mainly study the global structure of the quaternion Bernoulli equations × q   = aq + bq n for q Î \mathbb H , the quaternion field and also some other form of cubic quaternion differential equations. By using the Liouvillian theorem of integrability and the topological characterization of 2–dimensional torus: orientable compact connected surface of genus one, we prove that the quaternion Bernoulli equations may have invariant tori, which possesses a full Lebesgue measure subset of \mathbb H . Moreover, if n  = 2 all the invariant tori are full of periodic orbits; if n  = 3 there are infinitely many invariant tori fulfilling periodic orbits and also infinitely many invariant ones fulfilling dense orbits. Content Type Journal Article Pages 1-16 DOI 10.1007/s00220-011-1196-y Authors Xiang Zhang, Department of Mathematics, Shanghai Jiaotong University, Shanghai, 200240 P. R. China Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    A BV algebra and a QP-structure of the degree 3 is formulated. A QP-structure of degree 3 gives rise to Lie algebroids up to homotopy and its algebraic and geometric structure is analyzed. A new algebroid is constructed, which derives a new topological field theory in 4 dimensions by the AKSZ construction. Content Type Journal Article Pages 1-14 DOI 10.1007/s00220-011-1194-0 Authors Noriaki Ikeda, Maskawa Institute for Science and Culture, Kyoto Sangyo University, Kyoto, 603-8555 Japan Kyousuke Uchino, Department of Mathematics, Tokyo University of Science, Wakamiya 26, Shinjyuku, Tokyo, Japan Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We study the evolution of sharp fronts for the Surface Quasi-Geostrophic equation in the context of analytic functions. We showed that, even though the equation contains operators of order higher than 1, by carefully studying the evolution of the second derivatives it can be adapted to fit an abstract version of the Cauchy-Kowaleski Theorem. Content Type Journal Article Pages 1-28 DOI 10.1007/s00220-011-1190-4 Authors Charles Fefferman, Mathematics Department, Princeton University, Princeton, NJ 08544, USA José L. Rodrigo, Mathematics Department, Warwick University, Coventry, CV4 7AL UK Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to “light modes”, i.e. the elements of zero conformal weight, of the semi-infinite (BRST) cohomology complex of the Virasoro algebra associated with vertex operator algebra (VOA) with a formal parameter. We also construct a certain deformation of the BRST differential parametrized by a constant two-component tensor, such that it leads to the deformation of the A ∞ -subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the category of VOA with a formal parameter to the category of A ∞ -algebras. The associated generalized Maurer-Cartan equation gives the analogue of the Yang-Mills equation for a wide class of VOAs. Applying this construction to an example of VOA generated by β - γ systems, we find a remarkable relation between the Courant algebroid and the homotopy algebra of the Yang-Mills theory. Content Type Journal Article Pages 1-29 DOI 10.1007/s00220-011-1206-0 Authors Anton M. Zeitlin, Department of Mathematics, Yale University, 442 Dunham Lab, 10 Hillhouse Ave., New Haven, CT 06511, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements H xy , indexed by x , y Î L Ì \mathbb Z d , are independent, uniformly distributed random variables if \lvert x - y \rvert is less than the band width W , and zero otherwise. 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. All results are uniform in the size \lvert L \rvert of the matrix. Content Type Journal Article Pages 1-46 DOI 10.1007/s00220-011-1204-2 Authors László Erdős, Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany Antti Knowles, Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    A fundamental two-fluid model for describing dynamics of a plasma is the Euler-Poisson system, in which compressible ion and electron fluids interact with their self-consistent electrostatic force. Global smooth electron dynamics were constructed in Guo (Commun Math Phys 195:249–265, 1998 ) due to dispersive effect of the electric field. In this paper, we construct global smooth irrotational solutions with small amplitude for ion dynamics in the Euler-Poisson system. Content Type Journal Article Pages 1-37 DOI 10.1007/s00220-011-1193-1 Authors Yan Guo, Division of Applied Mathematics, Brown University, Providence, RI 02912, USA Benoit Pausader, Mathematics Department, Brown University, Providence, RI 02912, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    In a previous paper, we showed how one can obtain from the action of a locally compact quantum group on a type I -factor a possibly new locally compact quantum group. In another paper, we applied this construction method to the action of quantum SU (2) on the standard Podleś sphere to obtain Woronowicz’s quantum ~ E   (2) . In this paper, we will apply this technique to the action of quantum SU (2) on the quantum projective plane (whose associated von Neumann algebra is indeed a type I -factor). The locally compact quantum group which then comes out at the other side turns out to be the extended SU (1, 1) quantum group, as constructed by Koelink and Kustermans. We also show that there exists a (non-trivial) quantum groupoid which has at its corners (the duals of) the three quantum groups mentioned above. Content Type Journal Article Pages 1-42 DOI 10.1007/s00220-011-1208-y Authors Kenny De Commer, Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Consider a general strictly hyperbolic, quasilinear system, in one space dimesion u t + A ( u ) u x =0,                  (1) where u ® A ( u ), u Î W Ì \mathbb R N , is a smooth matrix-valued map. Given an initial datum u (0, ·) with small total variation, let u ( t , ·) be the corresponding (unique) vanishing viscosity solution of (1) obtained as a limit of solutions to the viscous parabolic approximation u t  +  A ( u ) u x  =  μ u xx , as  μ → 0. For every T  ≥ 0, we prove the a-priori bound || u e ( T ,·) - u ( T ,·) || \mathbb L 1 = o (1)· Ö   e     | log e |                   (1) for an approximate solution u e of (1) constructed by the Glimm scheme, with mesh size D x = D t = e , and with a suitable choice of the sampling sequence. This result provides for general hyperbolic systems the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems of conservation laws u t  +  F ( u ) x = 0 satisfying the classical Lax or Liu assumptions on the eigenvalues λ k ( u ) and on the eigenvectors r k ( u ) of the Jacobian matrix A ( u ) =  DF ( u ). The estimate (2) is obtained introducing a new wave interaction functional with a cubic term that controls the nonlinear coupling of waves of the same family and at the same time decreases at interactions by a quantity that is of the same order of the product of the wave strength times the change in the wave speeds. This is precisely the type of errors arising in a wave tracing analysis of the Glimm scheme, which is crucial to control in order to achieve an accurate estimate of the convergence rate as (2). Content Type Journal Article Pages 1-50 DOI 10.1007/s00220-010-1178-5 Authors Fabio Ancona, Dipartimento di Matematica Pura ed Applicata, Universita di Padova, Via Trieste 63, 35121 Padova, Italy Andrea Marson, Dipartimento di Matematica Pura ed Applicata, Universita di Padova, Via Trieste 63, 35121 Padova, Italy Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare elements of the class, and unstable in most cases. Anyhow, it is linearly stable (all orbits of the linearized system are bounded) and no motion is asymptotic in the past, namely no non-constant solution has the equilibrium as limit point as time goes to minus infinity. In the unstable cases, there is a sequence of initial data which converges to the equilibrium point whose corresponding solutions are unbounded and the motion is slow. So instability is quite weak and perhaps no such explicit examples of instability are known in the literature. The stable cases are also interesting since the level sets of the 2 first integrals independent and in involution keep being non-compact and stability is related to the isochronous periodicity of all orbits near the equilibrium point and the existence of a further first integral. Hopefully, these superintegrable Hamiltonian systems will deserve further research. Content Type Journal Article Pages 1-15 DOI 10.1007/s00220-011-1198-9 Authors Gaetano Zampieri, Università di Verona, Dipartimento di Informatica, Strada Le Grazie, 15, I-37134 Verona, Italy Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We give a geometric nonblow-up criterion on the direction of the vorticity for the three dimensional Navier-Stokes flow whose initial data is just bounded and may have infinite energy. We prove that under a restriction on behavior in time (type I condition) the solution does not blow up if the vorticity direction is uniformly continuous at the place where the vorticity magnitude is large. This improves the regularity condition for the vorticity direction first introduced by P. Constantin and C. Fefferman (1993) for finite energy weak solution. Our method is based on a simple blow-up argument which says that the situation looks like two-dimensional under continuity of the vorticity direction. We also discuss boundary value problems. Content Type Journal Article Pages 1-12 DOI 10.1007/s00220-011-1197-x Authors Yoshikazu Giga, Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Meguro, Tokyo, 153-8914 Japan Hideyuki Miura, Department of Mathematics, Osaka University, Machikaneyamacho 1-1, Toyonaka, Osaka, 560-0043 Japan Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We prove the asymptotic stability of the moving kinks for the nonlinear relativistic wave equations in one space dimension with a Ginzburg-Landau potential: starting in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of a uniformly moving kink and dispersive part described by the free Klein-Gordon equation. The remainder decays in a global energy norm. Our recent results on the weighted energy decay for the Klein-Gordon equations play a crucial role in the proofs. Content Type Journal Article Pages 1-28 DOI 10.1007/s00220-010-1184-7 Authors E. A. Kopylova, Institute for Information Transmission Problems RAS, B.Karetnyi 19, GSP-4, Moscow, 101447 Russia A. I. Komech, Institute for Information Transmission Problems RAS, B.Karetnyi 19, GSP-4, Moscow, 101447 Russia Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    We describe a method for comparing the spectra of two real-analytic families, ( a t ) and ( q t ), of quadratic forms that both degenerate as a positive parameter t tends to zero. We suppose that the family ( a t ) is amenable to ‘separation of variables’ and that each eigenspace of a t is 1-dimensional for some t . We show that if ( q t ) is asymptotic to ( a t ) at first order as t → 0, then the eigenspaces of ( q t ) are also 1-dimensional for all but countably many t . As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian acting on Dirichlet functions is 1-dimensional. Content Type Journal Article Pages 1-54 DOI 10.1007/s00220-010-1185-6 Authors Luc Hillairet, Laboratoire de Mathématiques Jean Leray, UMR 6629, Université de Nantes, 2, rue de la Houssinière, 44322 Nantes Cedex 3, France Chris Judge, Department of Mathematics, Indiana University, Bloomington, IN 47405, USA Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616

Description:    Let ℓ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let H be the half plane to the left of ℓ . We consider lozenge tilings of H that have a triangular gap of side-length two and in which ℓ is a free boundary — i.e., tiles are allowed to protrude out half-way across ℓ . We prove that the correlation function of this gap near the free boundary has asymptotics \frac14 p r , r → ∞, where r is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images. Content Type Journal Article Pages 1-37 DOI 10.1007/s00220-010-1186-5 Authors M. Ciucu, Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, USA C. Krattenthaler, Fakultät für Mathematik der Universität Wien, Nordbergstraße 15, A-1090 Wien, Austria Journal Communications in Mathematical Physics Online ISSN 1432-0916 Print ISSN 0010-3616