Authors:Alexis Drouot Pages: 757 - 806 Abstract: Abstract We present a proof of the existence of the Hawking radiation for massive bosons in the Schwarzschild--de Sitter metric. It provides estimates for the rates of decay of the initial quantum state to the Hawking thermal state. The arguments in the proof include a construction of radiation fields by conformal scattering theory; a semiclassical interpretation of the blueshift effect; the use of a WKB parametrix near the surface of a collapsing star. The proof does not rely on the spherical symmetry of the spacetime. PubDate: 2017-03-01 DOI: 10.1007/s00023-016-0509-2 Issue No:Vol. 18, No. 3 (2017)

Authors:Nicolò Drago; Thomas-Paul Hack; Nicola Pinamonti Pages: 807 - 868 Abstract: Abstract The principle of perturbative agreement, as introduced by Hollands and Wald, is a renormalization condition in quantum field theory on curved spacetimes. This principle states that the perturbative and exact constructions of a field theoretic model given by the sum of a free and an exactly tractable interaction Lagrangian should agree. We develop a proof of the validity of this principle in the case of scalar fields and quadratic interactions without derivatives, which differs in strategy from the one given by Hollands and Wald for the case of quadratic interactions encoding a change of metric. Thereby, we profit from the observation that, in the case of quadratic interactions, the composition of the inverse classical Møller map and the quantum Møller map is a contraction exponential of a particular type. Afterwards, we prove a generalisation of the principle of perturbative agreement and show that considering an arbitrary quadratic contribution of a general interaction either as part of the free theory or as part of the perturbation gives equivalent results. Motivated by the thermal mass idea, we use our findings to extend the construction of massive interacting thermal equilibrium states in Minkowski spacetime developed by Fredenhagen and Lindner to the massless case. In passing, we also prove a property of the construction of Fredenhagen and Lindner which was conjectured by these authors. PubDate: 2017-03-01 DOI: 10.1007/s00023-016-0521-6 Issue No:Vol. 18, No. 3 (2017)

Authors:Jan Dereziński; Serge Richard Pages: 869 - 928 Abstract: Abstract The paper is devoted to operators given formally by the expression $$\begin{aligned} -\partial _x^2+\left( \alpha -\frac{1}{4}\right) \frac{1}{x^{2}}. \end{aligned}$$ This expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real \(\alpha \) , or closed operator for complex \(\alpha \) , we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on \(L^2({\mathbb {R}}_+)\) , which we denote \(H_{m,\kappa }\) and \(H_0^\nu \) , with \(m^2=\alpha \) , \(-1<\mathrm{Re}(m)<1\) , and where \(\kappa ,\nu \in {\mathbb {C}}\cup \{\infty \}\) specify the boundary condition at 0. We study these operators using their explicit solvability in terms of Bessel-type functions and the Gamma function. In particular, we show that their point spectrum has a curious shape: a string of eigenvalues on a piece of a spiral. Their continuous spectrum is always \([0,\infty [\) . Restricted to their continuous spectrum, we diagonalize these operators using a generalization of the Hankel transformation. We also study their scattering theory. These operators are usually non-self-adjoint. Nevertheless, it is possible to use concepts typical for the self-adjoint case to study them. Let us also stress that \(-1<\mathrm{Re}(m)<1\) is the maximal region of parameters for which the operators \(H_{m,\kappa }\) can be defined within the framework of the Hilbert space \(L^2({\mathbb {R}}_+)\) . PubDate: 2017-03-01 DOI: 10.1007/s00023-016-0520-7 Issue No:Vol. 18, No. 3 (2017)

Authors:Pavel Exner; Stepan Manko Pages: 929 - 953 Abstract: Abstract We discuss spectral properties of a charged quantum particle confined to a chain graph consisting of an infinite array of rings under the influence of a magnetic field assuming a \({\delta}\) -coupling at the points where the rings touch. We start with the situation when the system has a translational symmetry and analyze spectral consequences of perturbations of various kind, such as a local change of the magnetic field, of the coupling constant, or of a ring circumference. A particular attention is paid to weak perturbations, both local and periodic; for the later, we prove a version of Saxon–Hutner conjecture. PubDate: 2017-03-01 DOI: 10.1007/s00023-016-0500-y Issue No:Vol. 18, No. 3 (2017)

Authors:Ivan Bardet Pages: 955 - 981 Abstract: Abstract In the study of open quantum systems modeled by a unitary evolution of a bipartite Hilbert space, we address the question of which parts of the environment can be said to have a “classical action” on the system, in the sense of acting as a classical stochastic process. Our method relies on the definition of the Environment Algebra, a relevant von Neumann algebra of the environment. With this algebra we define the classical parts of the environment and prove a decomposition between a maximal classical part and a quantum part. Then we investigate what other information can be obtained via this algebra, which leads us to define a more pertinent algebra: the Environment Action Algebra. This second algebra is linked to the minimal Stinespring representations induced by the unitary evolution on the system. Finally, in finite dimension we give a characterization of both algebras in terms of the spectrum of a certain completely positive map acting on the states of the environment. PubDate: 2017-03-01 DOI: 10.1007/s00023-016-0517-2 Issue No:Vol. 18, No. 3 (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:Nakao Hayashi; Pavel I. Naumkin Pages: 1025 - 1054 Abstract: Abstract We consider the Cauchy problem for the nonlinear Schrödinger equations of fractional order $$\left\{\begin{array}{l}i\partial _{t}u-2\left( -\partial _{x}^{2} \right)^{\frac{1}{4}} \, u=F\left( u\right) \\ u\left( 0,x\right) =u_{0} \left( x\right),\end{array}\right.$$ where \({F\left( u\right) }\) is the cubic nonlinearity $$F\left( u\right) =\lambda \left u\right ^{2}u$$ with \({\lambda \in \mathbf{R}}\) . We find the large time asymptotics of solutions to the Cauchy problem. We use the factorization technique similar to that developed for the Schrödinger equation. PubDate: 2017-03-01 DOI: 10.1007/s00023-016-0502-9 Issue No:Vol. 18, No. 3 (2017)

Authors:Abdellaziz Harrabi; Belgacem Rahal Pages: 1055 - 1094 Abstract: Abstract In this paper, we study the solutions of the triharmonic Lane–Emden equation $$\begin{aligned} -\Delta ^3 u= u ^{p-1}u,\quad \text{ in }\;\; \mathbb {R}^n, \quad \text{ with }\;\;n\ge 2\quad \text{ and }\quad p>1. \end{aligned}$$ As in Dávila et al. (Adv. Math. 258:240–285, 2014) and Farina (J. Math. Pures Appl. 87:537–561, 2007), we prove various Liouville-type theorems for smooth solutions under the assumption that they are stable or stable outside a compact set of \(\mathbb {R}^n\) . Again, following Dávila et al. (Adv. Math. 258:240–285, 2014), Hajlaoui et al. (On stable solutions of biharmonic prob- lem with polynomial growth. arXiv:1211.2223v2, 2012) and Wei and Ye (Math. Ann. 356:1599–1612, 2013), we first establish the standard integral estimates via stability property to derive the nonexistence results in the subcritical case by means of the Pohozaev identity. The supercritical case needs more involved analysis, motivated by the monotonicity formula established in Blatt (Monotonicity formulas for extrinsic triharmonic maps and the tri- harmonic Lane–Emden equation, 2014) (see also Luo et al., On the Triharmonic Lane–Emden Equation. arXiv:1607.04719, 2016), we then reduce the nonexistence of nontrivial entire solutions to that of nontrivial homogeneous solutions similarly to Dávila et al. (Adv. Math. 258:240–285, 2014). Through this approach, we give a complete classification of stable solutions and those which are stable outside a compact set of \(\mathbb {R}^n\) possibly unbounded and sign-changing. Inspired by Karageorgis (Nonlinearity 22:1653–1661, 2009), our analysis reveals a new critical exponent called the sixth-order Joseph–Lundgren exponent noted \(p_c(6,n)\) . Lastly, we give the explicit expression of \(p_c(6,n)\) . Our approach is less complicated and more transparent compared to Gazzola and Grunau (Math. Ann. 334:905–936, 2006) and Gazzola and Grunau (Polyharmonic boundary value problems. A monograph on positivity preserving and nonlinear higher order elliptic equations in bounded domains. Springer, New York, 2009) in terms of finding the explicit value of the fourth-Joseph–Lundgren exponent, \(p_c(4,n)\) . PubDate: 2017-03-01 DOI: 10.1007/s00023-016-0522-5 Issue No:Vol. 18, No. 3 (2017)

Authors:Zachary Bradshaw; Tai-Peng Tsai Pages: 1095 - 1119 Abstract: Abstract For any discretely self-similar, incompressible initial data which are arbitrarily large in weak \(L^3\) , we construct a forward discretely self-similar solution to the 3D Navier–Stokes equations in the whole space. This also gives a third construction of self-similar solutions for any \(-1\) -homogeneous initial data in weak \(L^3\) , improving those in JiaSverak and Šverák (Invent Math 196(1):233–265, 2014) and Korobkov and Tsai (Forward self-similar solutions of the Navier–Stokes equations in the half space, arXiv:1409.2516, 2016) for Hölder continuous data. Our method is based on a new, explicit a priori bound for the Leray equations. PubDate: 2017-03-01 DOI: 10.1007/s00023-016-0519-0 Issue No:Vol. 18, No. 3 (2017)

Authors:Teng Huang Pages: 623 - 641 Abstract: Abstract We consider an instanton, A, with L 2-bounded curvature F A on the cylindrical manifold \({Z=\mathbf{R} \times M}\) , where M is a closed Riemannian n-manifold, \({n \geq 4}\) . We assume M admits a smooth 3-form P and a smooth 4-form Q satisfy \({dP=4Q}\) and \({d\ast_{M}{Q}=(n-3)\ast_{M}P}\) . Manifolds with these forms include nearly Kähler 6-manifolds and nearly parallel G 2-manifolds in dimension 7. Then we can prove that the instanton must be a flat connection. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0503-8 Issue No:Vol. 18, No. 2 (2017)

Authors:Guan Huang Abstract: Abstract Consider a system of periodic pendulum lattice with analytic weak couplings: $$\begin{aligned} \ddot{x}_i+\sin x_i=-\epsilon \sum _{j=i-2}^i\partial _{x_i}\beta _{\alpha }(x_j,x_{j+1},x_{j+2}), \quad x_i=x_{i+N},\quad i\in \mathbb {Z}, \end{aligned}$$ where \(N\geqslant 3\) is an integer, \(\epsilon >0\) is a small parameter, and the function \(\beta _{\alpha }\) is an analytic function of a certain form. It is shown in this paper that for small enough \(\epsilon \) , the system admits motions such that the energy transfers between the pendulums in any predetermined order. PubDate: 2017-02-16 DOI: 10.1007/s00023-017-0561-6

Authors:Felix Finster; Moritz Reintjes Abstract: Abstract We give a nonperturbative construction of a distinguished state for the quantized Dirac field in Minkowski space in the presence of a time-dependent external field of the form of a plane electromagnetic wave. By explicit computation of the fermionic signature operator, it is shown that the Dirac operator has the strong mass oscillation property. We prove that the resulting fermionic projector state is a Hadamard state. PubDate: 2017-02-14 DOI: 10.1007/s00023-017-0557-2

Authors:Takuya Mine; Yuji Nomura 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: 2017-02-14 DOI: 10.1007/s00023-017-0559-0

Authors:Davide Gabrielli; Fabio Roncari Abstract: Abstract We call Alphabet model a generalization to N types of particles of the classic ABC model. We have particles of different types stochastically evolving on a one-dimensional lattice with an exchange dynamics. The rates of exchange are local, but under suitable conditions the dynamics is reversible with a Gibbsian-like invariant measure with long-range interactions. We discuss geometrically the conditions of reversibility on a ring that correspond to a gradient condition on the graph of configurations or equivalently to a divergence-free condition on a graph structure associated with the types of particles. We show that much of the information on the interactions between particles can be encoded in associated Tournaments that are a special class of oriented directed graphs. In particular we show that the interactions of reversible models are corresponding to strongly connected tournaments. The possible minimizers of the energies are in correspondence with the Hamiltonian cycles of the tournaments. We can then determine how many and which are the possible minimizers of the energy looking at the structure of the associated tournament. As a by-product we obtain a probabilistic proof of a classic Theorem of Camion (C R Acad Sci Paris 249: 2151–2152, 1959) on the existence of Hamiltonian cycles for strongly connected tournaments. Using these results, we obtain in the case of an equal number of k types of particles new representations of the Hamiltonians in terms of translation invariant k-body long range interactions. We show that when \(k=3,4\) the minimizer of the energy is always unique up to translations. Starting from the case \(k=5\) , it is possible to have more than one minimizer. In particular, it is possible to have minimizers for which particles of the same type are not joined together in single clusters. PubDate: 2017-02-13 DOI: 10.1007/s00023-017-0558-1

Authors:Jinho Baik; Ji Oon Lee Abstract: Abstract We consider a spherical spin system with pure 2-spin spherical Sherrington–Kirkpatrick Hamiltonian with ferromagnetic Curie–Weiss interaction. The system shows a two-dimensional phase transition with respect to the temperature and the coupling constant. We compute the limiting distributions of the free energy for all parameters away from the critical values. The zero temperature case corresponds to the well-known phase transition of the largest eigenvalue of a rank 1 spiked random symmetric matrix. As an intermediate step, we establish a central limit theorem for the linear statistics of rank 1 spiked random symmetric matrices. PubDate: 2017-02-11 DOI: 10.1007/s00023-017-0562-5

Authors:Rafael D. Benguria; Søren Fournais; Edgardo Stockmeyer; Hanne Van Den Bosch 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: 2017-02-06 DOI: 10.1007/s00023-017-0554-5

Authors:Ivan P. Costa e Silva; José Luis Flores Abstract: Abstract We study the interplay between the global causal and geometric structures of a spacetime (M, g) and the features of a given smooth \({\mathbb {R}}\) -action \(\rho \) on M whose orbits are all causal curves, building on classic results about Lie group actions on manifolds described by Palais (Ann Math 73:295–323, 1961). Although the dynamics of such an action can be very hard to describe in general, simple restrictions on the causal structure of (M, g) can simplify this dynamics dramatically. In the first part of this paper, we prove that \(\rho \) is free and proper (so that M splits topologically) provided that (M, g) is strongly causal and \(\rho \) does not have what we call weakly ancestral pairs, a notion which admits a natural interpretation in terms of “cosmic censorship.” Accordingly, such condition holds automatically if (M, g) is globally hyperbolic. We also prove that M splits topologically if (M, g) is strongly causal and \(\rho \) is the flow of a complete conformal Killing null vector field. In the second part, we investigate the class of Brinkmann spacetimes, which can be regarded as null analogues of stationary spacetimes in which \(\rho \) is the flow of a complete parallel null vector field. Inspired by the geometric characterization of stationary spacetimes in terms of standard stationary ones (Javaloyes and Sánchez in Class Quantum Gravity 25:168001, 2008), we obtain an analogous geometric characterization of when a Brinkmann spacetime is isometric to a standard Brinkmann spacetime. This result naturally leads us to discuss a conjectural null analogue for Ricci-flat four-dimensional Brinkmann spacetimes of a celebrated rigidity theorem by Anderson (Ann Henri Poincaré 1:977–994, 2000) and highlight its relation with a long-standing conjecture by Ehlers and Kundt (Gravitation: an introduction to current research. Wiley, New York, pp 49–101, 1962). PubDate: 2017-02-06 DOI: 10.1007/s00023-017-0551-8

Authors:Po-Ning Chen; Pei-Ken Hung; Mu-Tao Wang; Shing-Tung Yau Abstract: Abstract We study the space of Killing fields on the four dimensional AdS spacetime \(AdS^{3,1}\) . Two subsets \({\mathcal {S}}\) and \({\mathcal {O}}\) are identified: \({\mathcal {S}}\) (the spinor Killing fields) is constructed from imaginary Killing spinors, and \({\mathcal {O}}\) (the observer Killing fields) consists of all hypersurface orthogonal, future timelike unit Killing fields. When the cosmology constant vanishes, or in the Minkowski spacetime case, these two subsets have the same convex hull in the space of Killing fields. In presence of the cosmology constant, the convex hull of \( {\mathcal {O}}\) is properly contained in that of \({\mathcal {S}}\) . This leads to two different notions of energy for an asymptotically AdS spacetime, the spinor energy and the observer energy. Chruściel et al. (J High Energy Phys 2006(11):084, 2006) proved the positivity of the spinor energy and derived important consequences among the related conserved quantities. We show that the positivity of the observer energy follows from the positivity of the spinor energy. A new notion called the “rest mass” of an asymptotically AdS spacetime is then defined by minimizing the observer energy and is shown to be evaluated in terms of the adjoint representation of the Lie algebra of Killing fields. It is proved that the rest mass has the desirable rigidity property that characterizes the AdS spacetime. PubDate: 2017-02-02 DOI: 10.1007/s00023-017-0555-4

Authors:Oleg Safronov 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: 2017-02-02 DOI: 10.1007/s00023-017-0553-6