Abstract: Abstract
As an inclusive
\({(1,3)\ni p}\)
—extension of Bray–Miao’s Theorem 1 and Corollary 1 (Invent Math 172:459–475, 2008) for p = 2, this note presents a sharp isoperimetric inequality for the p-harmonic capacity of a surface in the complete, smooth, asymptotically flat 3-manifold with non-negative scalar curvature, and then an optimal Riemannian Penrose type inequality linking the ADM/total mass and the p-harmonic capacity by means of the deficit of Willmore’s energy. Even in the Euclidean 3-space, the discovered result for
\({p \not =2}\)
is new and non-trivial. PubDate: 2016-08-01

Abstract: Abstract
In this paper, we introduce new methods for solving the vacuum Einstein constraints equations: the first one is based on Schaefer’s fixed point theorem (known methods use Schauder’s fixed point theorem), while the second one uses the concept of half-continuity coupled with the introduction of local supersolutions. These methods allow to: unify some recent existence results, simplify many proofs (for instance, the one of the main theorems in Dahl et al., Duke Math J 161(14):2669–2697, 2012) and weaken the assumptions of many recent results. PubDate: 2016-08-01

Abstract: We prove pointwise in time decay estimates via an abstract conjugate operator method. This is then applied to a large class of dispersive equations. PubDate: 2016-08-01

Abstract: Abstract
We present a generalization of Minkowski’s classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi–Civita holonomies around each of the tetrahedron’s faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra within a unified framework. A new type of hyperbolic simplex is introduced in order for all the sectors encoded in the algebraic data to be covered. Generalizing the phase space of shapes associated to flat tetrahedra leads to group-valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant. This becomes manifest in light of their relation with the spin-network states of loop quantum gravity. This work therefore provides a bottom-up justification for the emergence of deformed gauge symmetries and quantum groups in covariant loop quantum gravity in the presence of a cosmological constant. PubDate: 2016-08-01

Abstract: Abstract
We consider perturbations of the semiclassical Schrödinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the perturbation, the solutions associated to initial data in a small spectral window become equidistributed in the semiclassical limit. As an application of our method, we also derive some properties of the quantum Loschmidt echo below and beyond the Ehrenfest time for initial data in a small spectral window. PubDate: 2016-08-01

Abstract: Abstract
We study the spectral properties of Schrödinger operators on perturbed lattices. We shall prove the non-existence or the discreteness of embedded eigenvalues, the limiting absorption principle for the resolvent, construct a spectral representation, and define the S-matrix. Our theory covers the square, triangular, diamond, Kagome lattices, as well as the ladder, the graphite and the subdivision of square lattice. PubDate: 2016-08-01

Abstract: Abstract
We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. In Martinez and Sordoni (Mem AMS 936, 2009) such a case is also studied but their reduced Hamiltonian includes the vector potential terms. In this paper, using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms of the nucleus. Using the reduced evolution, we also obtain the asymptotic expansion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constant magnetic fields. PubDate: 2016-08-01

Abstract: Abstract
Differential equations with infinitely many derivatives, sometimes also referred to as “nonlocal” differential equations, appear frequently in branches of modern physics such as string theory, gravitation and cosmology. We properly interpret and solve linear equations in this class with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous studies of nonlocal equations via Laplace and Fourier transforms, see for instance (Barnaby and Kamran, J High Energy Phys 02:40, 2008; Górka et al., Class Quantum Gravity 29:065017, 2012; Górka et al., Ann Henri Poincaré 14:947–966, 2013). We reconsider “generalized” initial value problems within the present approach and we disprove various conjectures found in modern physics literature. We illustrate various analytic phenomena that can occur with concrete examples, and we also treat efficient implementations of the theory. PubDate: 2016-08-01

Abstract: Abstract
This note provides a C
*-algebraic framework for supersymmetry. Particularly, we consider fermion lattice models satisfying the simplest supersymmetry relation. Namely, we discuss a restricted sense of supersymmetry without a boson field involved. We construct general supersymmetric C
*-dynamics in terms of a superderivation and a one-parameter group of automorphisms on the CAR algebra. (We do not introduce Grassmann numbers into our formalism.) We show several basic properties of superderivations on the fermion lattice system. Among others, we establish that superderivations defined on the strictly local algebra are norm-closable. We show a criterion of superderivations on the fermion lattice system for being nilpotent. This criterion can be easily checked and hence yields new supersymmetric fermion lattice models. PubDate: 2016-08-01

Abstract: Abstract
We study vacuum polarisation effects of a Dirac field coupled to an external scalar field and derive a semi-classical expansion of the regularised vacuum energy. The leading order of this expansion is given by a classical formula due to Chin, Lee-Wick and Walecka, for which our result provides the first rigorous proof. We then discuss applications to the non-relativistic large-coupling limit of an interacting system, and to the stability of homogeneous systems. PubDate: 2016-08-01

Abstract: Abstract
We consider the many-body quantum dynamics of systems of bosons interacting through a two-body potential
\({N^{3\beta-1} V (N^\beta x)}\)
, scaling with the number of particles N. For
\({0 < \beta < 1}\)
, we obtain a norm-approximation of the evolution of an appropriate class of data on the Fock space. To this end, we need to correct the evolution of the condensate described by the one-particle nonlinear Schrödinger equation by means of a fluctuation dynamics, governed by a quadratic generator. PubDate: 2016-07-20

Abstract: Abstract
We show that all generalized Pollicott–Ruelle resonant states of a topologically transitive C
∞ Anosov flow with an arbitrary C
∞ potential have full support. PubDate: 2016-07-16

Abstract: Abstract
This study is devoted to the cubic nonlinear Schrödinger equation in a two-dimensional waveguide with shrinking cross section of order
\({\varepsilon}\)
. For the Cauchy data living essentially on the first mode of the transverse Laplacian, we provide a tensorial approximation of the solution
\({\psi^{\varepsilon}}\)
in the limit
\({\varepsilon \to 0}\)
, with an estimate of the approximation error, and derive a limiting nonlinear Schrödinger equation in dimension one. If the Cauchy data
\({\psi^{\varepsilon}_0}\)
have a uniformly bounded energy, then it is a bounded sequence in
\({\mathsf{H}^1}\)
, and we show that the approximation is of order
\({\mathcal{O}(\sqrt{\varepsilon})}\)
. If we assume that
\({\psi^{\varepsilon}_0}\)
is bounded in the graph norm of the Hamiltonian, then it is a bounded sequence in
\({\mathsf{H}^{2}}\)
, and we show that the approximation error is of order
\({\mathcal{O}(\varepsilon)}\)
. PubDate: 2016-07-16

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: 2016-07-15

Abstract: Abstract
We consider the focusing nonlinear cubic Klein–Gordon equation in three spatial dimensions (NLKG). By the classical result of Payne and Sattinger (Israel J Math 22(3–4):273–303, 1975), one can distinguish global existence and blow-up for real-valued initial data with energy less than that of the ground state, by a variational characterization. Following the idea of Kenig and Merle (Acta Math 201(2):147–212, 2008), Ibrahim, Masmoudi, and Nakanishi (Anal PDE 4(3):405–460, 2011) proved scattering of real-valued solutions in the region of global existence. We will extend their classification to complex-valued solutions below the ground-state standing waves using the charge (see Theorem 1.1). We apply their method to obtain the scattering result. The difference is that they use only the energy, but we need to control both the energy and the charge. On the other hand, we cannot use their proof, which is based on Payne and Sattinger (Israel J Math 22(3–4):273–303, 1975), to obtain the blow-up result. Therefore, we combine the idea of Holmer and Roudenko (Commun Partial Differ Equ 35(5):878–905, 2010), who proved a similar classification for the nonlinear Schrödinger equation, with the idea of Ohta and Todorova (SIAM J Math Anal 38(6):1912–1931, 2007). Moreover, we extend the classification using not only the energy and the charge, but also the momentum (see Theorem 1.3). Due to these extensions, we can classify the solutions which have large energy. PubDate: 2016-07-15

Abstract: Abstract
The initial boundary value problem for a damped fourth-order wave equation with exponential growth nonlinearity is investigated in four space dimensions. In the defocusing case, global well-posedness, scattering and exponential decay are established. In the focusing sign, existence of ground state, instability of standing waves and blow-up results are obtained. PubDate: 2016-07-14

Abstract: Abstract
We study damped geodesic motion of a particle of mass m on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as
\({m \to 0}\)
, its solutions converge to solutions of a limiting equation which includes a noise-induced drift term. A very special case of the main result presents Brownian motion on the manifold as a limit of inertial systems. PubDate: 2016-07-11

Abstract: Abstract
The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses ≤ 1. The process stabilizes almost surely if m < 1 and it almost surely does not stabilize if m > 1, where m is the mean mass per vertex. The main result of this paper is that in the critical case m = 1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete bi-Laplacian Gaussian field. PubDate: 2016-07-01

Abstract: Abstract
We place the hyperbolic quantum Ruijsenaars–Schneider system with an exponential Morse term on a lattice and diagonalize the resulting n-particle model by means of multivariate continuous dual q-Hahn polynomials that arise as a parameter reduction of the Macdonald–Koornwinder polynomials. This allows to compute the n-particle scattering operator, to identify the bispectral dual system, and to confirm the quantum integrability in a Hilbert space setup. PubDate: 2016-07-01

Abstract: Abstract
We consider commuting operators obtained by quantization of Hamiltonians of the Hopf (aka dispersionless KdV) hierarchy. Such operators naturally arise in the setting of Symplectic Field Theory (SFT). A complete set of common eigenvectors of these operators is given by Schur polynomials. We use this result for computing the SFT potential of a disk. PubDate: 2016-07-01