Authors:Alexis Drouot Pages: 757 - 806 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: 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: 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: 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: 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: 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: 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: 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: 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: 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:Martin Fraas; Lisa Hänggli 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: 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: 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: 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: 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: 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: 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

Authors:Piotr Tourkine Abstract: In this work, we argue that the \(\alpha '\rightarrow 0\) limit of closed string theory scattering amplitudes is a tropical limit. The motivation is to develop a technology to systematize the extraction of Feynman graphs from string theory amplitudes at higher genus. An important technical input from tropical geometry is the use of tropical theta functions with characteristics to rigorously derive the worldline limit of the worldsheet propagator. This enables us to perform a non-trivial computation at two loops: we derive the tropical form of the integrand of the genus-two four-graviton type II string amplitude, which matches the direct field theory computations. At the mathematical level, this limit is an implementation of the correspondence between the moduli space of Riemann surfaces and the tropical moduli space. PubDate: 2017-02-25 DOI: 10.1007/s00023-017-0560-7

Authors:Giuseppe De Nittis; Max Lein Abstract: In this work, we present a novel approach to the ray optics limit: we rewrite the dynamical Maxwell equations in Schrödinger form and prove Egorov-type theorems, a robust semiclassical technique. We implement this scheme for periodic light conductors, photonic crystals, thereby making the quantum-light analogy between semiclassics for the Bloch electron and ray optics in photonic crystals rigorous. One major conceptual difference between the two theories, though, is that electromagnetic fields are real, and hence, we need to add one step in the derivation to reduce it to a single-band problem. Our main results, Theorem 3.7 and Corollary 3.9, give a ray optics limit for quadratic observables and, among others, apply to local averages of energy density, the Poynting vector and the Maxwell stress tensor. Ours is the first rigorous derivation of ray optics equations which include all subleading-order terms, some of which are also new to the physics literature. The ray optics limit we prove applies to photonic crystals of any topological class. PubDate: 2017-02-24 DOI: 10.1007/s00023-017-0552-7

Authors:Tristan Benoist; Clément Pellegrini; Francesco Ticozzi Abstract: We study the stability of quantum pure states and, more generally, subspaces for stochastic dynamics that describe continuously monitored systems. We show that the target subspace is almost surely invariant if and only if it is invariant for the average evolution and that the same equivalence holds for the global asymptotic stability. Moreover, we prove that a strict linear Lyapunov function for the average evolution always exists, and latter can be used to derive sharp bounds on the Lyapunov exponents of the associated semigroup. Nonetheless, we also show that taking into account the measurements can lead to an improved bound on stability rate for the stochastic, non-averaged dynamics. We discuss explicit examples where the almost sure stability rate can be made arbitrary large while the average one stays constant. PubDate: 2017-02-23 DOI: 10.1007/s00023-017-0556-3