Authors:Luigi Cantini Pages: 1121 - 1151 Abstract: Abstract In this paper, we analyze the steady state of the asymmetric simple exclusion process with open boundaries and second class particles by deforming it through the introduction of spectral parameters. The (unnormalized) probabilities of the particle configurations get promoted to Laurent polynomials in the spectral parameters and are constructed in terms of non-symmetric Koornwinder polynomials. In particular, we show that the partition function coincides with a symmetric Macdonald–Koornwinder polynomial. As an outcome, we compute the steady current and the average density of first class particles. PubDate: 2017-04-01 DOI: 10.1007/s00023-016-0540-3 Issue No:Vol. 18, No. 4 (2017)

Authors:Daniel Puzzuoli; John Watrous Pages: 1153 - 1184 Abstract: Abstract Single-shot quantum channel discrimination is a fundamental task in quantum information theory. It is well known that entanglement with an ancillary system can help in this task, and, furthermore, that an ancilla with the same dimension as the input of the channels is always sufficient for optimal discrimination of two channels. A natural question to ask is whether the same holds true for the output dimension. That is, in cases when the output dimension of the channels is (possibly much) smaller than the input dimension, is an ancilla with dimension equal to the output dimension always sufficient for optimal discrimination? We show that the answer to this question is “no” by construction of a family of counterexamples. This family contains instances with arbitrary finite gap between the input and output dimensions, and still has the property that in every case, for optimal discrimination, it is necessary to use an ancilla with dimension equal to that of the input. The proof relies on a characterization of all operators on the trace norm unit sphere that maximize entanglement negativity. In the case of density operators, we generalize this characterization to a broad class of entanglement measures, which we call weak entanglement measures. This characterization allows us to conclude that a quantum channel is reversible if and only if it preserves entanglement as measured by any weak entanglement measure, with the structure of maximally entangled states being equivalent to the structure of reversible maps via the Choi isomorphism. We also include alternate proofs of other known characterizations of channel reversibility. PubDate: 2017-04-01 DOI: 10.1007/s00023-016-0537-y Issue No:Vol. 18, No. 4 (2017)

Authors:Dmitry Pelinovsky; Guido Schneider Pages: 1185 - 1211 Abstract: Abstract The nonlinear Schrödinger (NLS) equation is considered on a periodic graph subject to the Kirchhoff boundary conditions. Bifurcations of standing localized waves for frequencies lying below the bottom of the linear spectrum of the associated stationary Schrödinger equation are considered by using analysis of two-dimensional discrete maps near hyperbolic fixed points. We prove the existence of two distinct families of small-amplitude standing localized waves, which are symmetric about the two symmetry points of the periodic graph. We also prove properties of the two families, in particular, positivity and exponential decay. The asymptotic reduction of the two-dimensional discrete map to the stationary NLS equation on an infinite line is discussed in the context of the homogenization of the NLS equation on the periodic graph. PubDate: 2017-04-01 DOI: 10.1007/s00023-016-0536-z Issue No:Vol. 18, No. 4 (2017)

Authors:Xuecheng Wang Pages: 1213 - 1267 Abstract: Abstract We establish the global existence and the asymptotic behavior for the 2D incompressible isotropic elastodynamics for sufficiently small, smooth initial data in the Eulerian coordinates formulation. The main tools used to derive the main results are, on the one hand, a modified energy method to derive the energy estimate and, on the other hand, a Fourier transform method with a suitable choice of Z-norm to derive the sharp \(L^\infty \) -estimate. We mention that the global existence of the same system but in the Lagrangian coordinates formulation was recently obtained by Lei (Global well-posedness of incompressible Elastodynamics in 2D, 2014). Our goal is to improve the understanding of the behavior of solutions. Also, we present a different approach to study 2D nonlinear wave equations from the point of view in frequency space. PubDate: 2017-04-01 DOI: 10.1007/s00023-016-0538-x Issue No:Vol. 18, No. 4 (2017)

Authors:Michael Goldberg; William R. Green Pages: 1269 - 1288 Abstract: Abstract Let \(H=-\Delta +V\) be a Schrödinger operator on \(L^2(\mathbb {R}^4)\) with real-valued potential V, and let \(H_0=-\Delta \) . If V has sufficient pointwise decay, the wave operators \(W_{\pm }=s-\lim _{t\rightarrow \pm \infty } e^{itH}e^{-itH_0}\) are known to be bounded on \(L^p(\mathbb {R}^4)\) for all \(1\le p\le \infty \) if zero is not an eigenvalue or resonance, and on \(\frac{4}{3}<p<4\) if zero is an eigenvalue but not a resonance. We show that in the latter case, the wave operators are also bounded on \(L^p(\mathbb {R}^4)\) for \(1\le p\le \frac{4}{3}\) by direct examination of the integral kernel of the leading terms. Furthermore, if \(\int _{\mathbb {R}^4} xV(x) \psi (x) \, dx=0\) for all zero energy eigenfunctions \(\psi \) , then the wave operators are bounded on \(L^p\) for \(1 \le p<\infty \) . PubDate: 2017-04-01 DOI: 10.1007/s00023-016-0534-1 Issue No:Vol. 18, No. 4 (2017)

Authors:Jussi Behrndt; Rupert L. Frank; Christian Kühn; Vladimir Lotoreichik; Jonathan Rohleder Pages: 1305 - 1347 Abstract: Abstract The main objective of this paper is to systematically develop a spectral and scattering theory for self-adjoint Schrödinger operators with \(\delta \) -interactions supported on closed curves in \(\mathbb {R}^3\) . We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive Schatten–von Neumann properties for the resolvent difference with the free Laplacian, and establish an explicit representation for the scattering matrix. PubDate: 2017-04-01 DOI: 10.1007/s00023-016-0532-3 Issue No:Vol. 18, No. 4 (2017)

Authors:Takuya Mine; Yuji Nomura Pages: 1349 - 1369 Abstract: Abstract We shall consider the Schrödinger operators on \(\mathbb {R}^2\) with random \(\delta \) magnetic fields. Under some mild conditions on the positions and the fluxes of the \(\delta \) -fields, we prove the spectrum coincides with \([0,\infty )\) and the integrated density of states (IDS) decays exponentially at the bottom of the spectrum (Lifshitz tail), by using the Hardy-type inequality by Laptev-Weidl (Oper Theory Adv Appl 108:299–305, 1999). We also give a lower bound for IDS at the bottom of the spectrum. PubDate: 2017-04-01 DOI: 10.1007/s00023-017-0559-0 Issue No:Vol. 18, No. 4 (2017)

Authors:Rafael D. Benguria; Søren Fournais; Edgardo Stockmeyer; Hanne Van Den Bosch Pages: 1371 - 1383 Abstract: Abstract We consider Dirac operators defined on planar domains. For a large class of boundary conditions, we give a direct proof of their self-adjointness in the Sobolev space \(H^1\) . PubDate: 2017-04-01 DOI: 10.1007/s00023-017-0554-5 Issue No:Vol. 18, No. 4 (2017)

Authors:Oleg Safronov Pages: 1385 - 1434 Abstract: Abstract We consider different Dirac operators with either electric or magnetic potentials satisfying conditions guaranteeing their decay at infinity. We prove that the absolutely continuous spectra of these operators cover the intervals \((-\infty ,-1]\) and \([1,\infty )\) . In particular, we prove an analogue of Simon’s conjecture for the magnetic Dirac operator. PubDate: 2017-04-01 DOI: 10.1007/s00023-017-0553-6 Issue No:Vol. 18, No. 4 (2017)

Authors:Marco Benini; Alexander Schenkel Pages: 1435 - 1464 Abstract: Abstract We develop an approach to construct Poisson algebras for non-linear scalar field theories that is based on the Cahiers topos model for synthetic differential geometry. In this framework, the solution space of the field equation carries a natural smooth structure and, following Zuckerman’s ideas, we can endow it with a presymplectic current. We formulate the Hamiltonian vector field equation in this setting and show that it selects a family of observables which forms a Poisson algebra. Our approach provides a clean splitting between geometric and algebraic aspects of the construction of a Poisson algebra, which are sufficient to guarantee existence, and analytical aspects that are crucial to analyze its properties. PubDate: 2017-04-01 DOI: 10.1007/s00023-016-0533-2 Issue No:Vol. 18, No. 4 (2017)

Authors:Etera R. Livine Pages: 1465 - 1491 Abstract: Abstract The Ponzano–Regge state-sum model provides a quantization of 3d gravity as a spin foam, providing a quantum amplitude to each 3d triangulation defined in terms of the 6j-symbol (from the spin-recoupling theory of \(\mathrm {SU}(2)\) representations). In this context, the invariance of the 6j-symbol under 4-1 Pachner moves, mathematically defined by the Biedenharn–Elliott identity, can be understood as the invariance of the Ponzano–Regge model under coarse-graining or equivalently as the invariance of the amplitudes under the Hamiltonian constraints. Here, we look at length and volume insertions in the Biedenharn–Elliott identity for the 6j-symbol, derived in some sense as higher derivatives of the original formula. This gives the behavior of these geometrical observables under coarse-graining. These new identities turn out to be related to the Biedenharn–Elliott identity for the q-deformed 6j-symbol and highlight that the q-deformation produces a cosmological constant term in the Hamiltonian constraints of 3d quantum gravity. PubDate: 2017-04-01 DOI: 10.1007/s00023-016-0535-0 Issue No:Vol. 18, No. 4 (2017)

Authors:András Vasy Pages: 983 - 1007 Abstract: Abstract We discuss positivity properties of certain‘distinguished propagators’, i.e., distinguished inverses of operators that frequently occur in scattering theory and wave propagation. We relate this to the work of Duistermaat and Hörmander on distinguished parametrices (approximate inverses), which has played a major role in quantum field theory on curved spacetimes recently. PubDate: 2017-03-01 DOI: 10.1007/s00023-016-0527-0 Issue No:Vol. 18, No. 3 (2017)

Authors:Sigmund Selberg; Daniel Oliveira da Silva Pages: 1009 - 1023 Abstract: Abstract We present lower bounds for the uniform radius of spatial analyticity of solutions to the Korteweg–de Vries equation, which improve earlier results due to Bona, Grujić and Kalisch. PubDate: 2017-03-01 DOI: 10.1007/s00023-016-0498-1 Issue No:Vol. 18, No. 3 (2017)

Authors:Martin Fraas; Lisa Hänggli Abstract: Abstract We consider a driven open system whose evolution is described by a Lindbladian. The Lindbladian is assumed to be dephasing and its Hamiltonian part to be given by the Landau–Zener Hamiltonian. We derive a formula for the transition probability which, unlike previous results, extends the Landau–Zener formula to open systems. PubDate: 2017-03-18 DOI: 10.1007/s00023-017-0567-0

Authors:Sergey Morozov; David Müller Abstract: Abstract Considering different self-adjoint realisations of positively projected massless Coulomb–Dirac operators we find out under which conditions any negative perturbation, however small, leads to emergence of negative spectrum. We also prove some weighted Lieb–Thirring estimates for negative eigenvalues of such operators. In the process we find explicit spectral representations for all self-adjoint realisations of massless Coulomb–Dirac operators on the half-line. PubDate: 2017-03-17 DOI: 10.1007/s00023-017-0570-5

Authors:Oran Gannot Abstract: Abstract This paper establishes the existence of quasinormal frequencies converging exponentially to the real axis for the Klein–Gordon equation on a Kerr–AdS spacetime when Dirichlet boundary conditions are imposed at the conformal boundary. The proof is adapted from results in Euclidean scattering about the existence of scattering poles generated by time-periodic approximate solutions to the wave equation. PubDate: 2017-03-16 DOI: 10.1007/s00023-017-0568-z

Authors:Fumio Hiai; Robert König; Marco Tomamichel Abstract: Abstract We show that recent multivariate generalizations of the Araki–Lieb–Thirring inequality and the Golden–Thompson inequality (Sutter et al. in Commun Math Phys, 2016. doi:10.1007/s00220-016-2778-5) for Schatten norms hold more generally for all unitarily invariant norms and certain variations thereof. The main technical contribution is a generalization of the concept of log-majorization which allows us to treat majorization with regard to logarithmic integral averages of vectors of singular values. PubDate: 2017-03-13 DOI: 10.1007/s00023-017-0569-y

Authors:Michał Eckstein; Tomasz Miller Abstract: Abstract Drawing from the theory of optimal transport we propose a rigorous notion of a causal relation for Borel probability measures on a given spacetime. To prepare the ground, we explore the borderland between Lorentzian geometry, topology and measure theory. We provide various characterisations of the proposed causal relation, which turn out to be equivalent if the underlying spacetime has a sufficiently robust causal structure. We also present the notion of the ‘Lorentz–Wasserstein distance’ and study its basic properties. Finally, we outline the possible applications of the developed formalism in both classical and quantum physics. PubDate: 2017-03-13 DOI: 10.1007/s00023-017-0566-1

Authors:Paul T. Allen; Iva Stavrov Allen Abstract: Abstract We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in Hölder norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data. PubDate: 2017-03-10 DOI: 10.1007/s00023-017-0565-2

Authors:Marzia Dalla Venezia; André Martinez Abstract: Abstract We study the widths of shape resonances for the semiclassical multidimensional Schrödinger operator, in the case where the frequency remains close to some value strictly larger than the bottom of the well. Under a condition on the behavior of the resonant state inside the well, we obtain an optimal lower bound for the widths. PubDate: 2017-03-01 DOI: 10.1007/s00023-017-0564-3