Authors:Yaiza Canzani; John A. Toth Pages: 3063 - 3087 Abstract: Abstract This note concerns the nodal sets of eigenfunctions of semiclassical Schrödinger operators acting on compact, smooth, Riemannian manifolds, with no boundary. In the case of real analytic surfaces, we obtain sharp upper bounds for the number of intersections of the zero sets of Schrödinger eigenfunctions with a fixed curve that lies inside the classically forbidden region. PubDate: 2016-11-01 DOI: 10.1007/s00023-016-0488-3 Issue No:Vol. 17, No. 11 (2016)

Abstract: Abstract We consider a system of N bosons confined to a thin waveguide, i.e. to a region of space within an \({\epsilon}\) -tube around a curve in \({\mathbb{R}^3}\) . We show that when taking simultaneously the NLS limit \({N \to \infty}\) and the limit of strong confinement \({\epsilon \to 0}\) , the time-evolution of such a system starting in a state close to a Bose–Einstein condensate is approximately captured by a non-linear Schrödinger equation in one dimension. The strength of the non-linearity in this Gross–Pitaevskii type equation depends on the shape of the cross-section of the waveguide, while the “bending” and the “twisting” of the waveguide contribute potential terms. Our analysis is based on an approach to mean-field limits developed by Pickl (On the time-dependent Gross–Pitaevskii-and Hartree equation. arXiv:0808.1178, 2008). PubDate: 2016-12-01 DOI: 10.1007/s00023-016-0487-4

Abstract: Abstract Given a family of self-adjoint operators \({(A_t)_{t \in T}}\) indexed by a parameter t in some topological space T, necessary and sufficient conditions are given for the spectrum \({\sigma(A_t)}\) to be Vietoris continuous with respect to t. Equivalently the boundaries and the gap edges are continuous in t. If (T, d) is a complete metric space with metric d, these conditions are extended to guarantee Hölder continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps. PubDate: 2016-12-01 DOI: 10.1007/s00023-016-0496-3

Abstract: Abstract We study a classical lattice dipole gas with low activity in dimension \({d \geq 3}\) . We investigate long distance properties by a renormalization group analysis. We prove that various correlation functions have an infinite volume limit. We also get estimates on the decay of correlation functions. PubDate: 2016-12-01 DOI: 10.1007/s00023-016-0495-4

Abstract: Abstract We give a constructive proof for the existence of a Bloch basis of rank \({N}\) which is both smooth (real analytic) and periodic with respect to its \({d}\) -dimensional quasi-momenta, when \({1\leq d\leq 2}\) and \({N\geq 1}\) . The constructed Bloch basis is conjugation symmetric when the underlying projection has this symmetry, hence the corresponding exponentially localized composite Wannier functions are real. In the second part of the paper, we show that by adding a weak, globally bounded but not necessarily constant magnetic field, the existence of a localized basis is preserved. PubDate: 2016-12-01 DOI: 10.1007/s00023-016-0489-2

Abstract: Abstract We study the asymptotic distribution of the resonances near the Landau levels \({\Lambda_q =(2q+1)b}\) , \({q \in \mathbb{N}}\) , of the Dirichlet (resp. Neumann, resp. Robin) realization in the exterior of a compact domain of \({\mathbb{R}^3}\) of the 3D Schrödinger operator with constant magnetic field of scalar intensity \({b > 0}\) . We investigate the corresponding resonance counting function and obtain the main asymptotic term. In particular, we prove the accumulation of resonances at the Landau levels and the existence of resonance-free sectors. In some cases, it provides the discreteness of the set of embedded eigenvalues near the Landau levels. PubDate: 2016-12-01 DOI: 10.1007/s00023-016-0497-2

Abstract: Abstract De Rham cohomology with spacelike compact and timelike compact supports has recently been noticed to be of importance for understanding the structure of classical and quantum Maxwell theory on curved spacetimes. Similarly, causally restricted cohomologies of different differential complexes play a similar role in other gauge theories. We introduce a method for computing these causally restricted cohomologies in terms of cohomologies with either compact or unrestricted supports. The calculation exploits the fact that the de Rham–d’Alembert wave operator can be extended to a chain map that is homotopic to zero and that its causal Green function fits into a convenient exact sequence. As a first application, we use the method on the de Rham complex, then also on the Calabi (or Killing–Riemann–Bianchi) complex, which appears in linearized gravity on constant curvature backgrounds. We also discuss applications to other complexes, as well as generalized causal structures and functoriality. PubDate: 2016-12-01 DOI: 10.1007/s00023-016-0481-x

Abstract: Abstract In the small noise regime, the average transition time between metastable states of a reversible diffusion process is described at the logarithmic scale by Arrhenius’ law. The Eyring–Kramers formula classically provides a subexponential prefactor to this large deviation estimate. For irreversible diffusion processes, the equivalent of Arrhenius’ law is given by the Freidlin–Wentzell theory. In this paper, we compute the associated prefactor and thereby generalise the Eyring–Kramers formula to irreversible diffusion processes. In our formula, the role of the potential is played by Freidlin–Wentzell’s quasipotential, and a correction depending on the non-Gibbsianness of the system along the minimum action paths is highlighted. Our study assumes some properties for the vector field: (1) attractors are isolated points, (2) the dynamics restricted to basin of attraction boundaries are attracted to single points (which are saddle-points of the vector field). We moreover assume that the minimum action paths that connect attractors to adjacent saddle-points (the instantons) have generic properties that are summarised in the conclusion. At a technical level, our derivation combines an exact computation for the first-order WKB expansion around the instanton and an exact computation of the first-order match asymptotics expansion close to the saddle-point. While the results are exact once a formal expansion is assumed, the validity of these asymptotic expansions remains to be proven. PubDate: 2016-12-01 DOI: 10.1007/s00023-016-0507-4

Abstract: Abstract Recently we introduced T-duality in the study of topological insulators, and used it to show that T-duality transforms the bulk–boundary homomorphism into a simpler restriction map in two dimensions. In this paper, we partially generalize these results to higher dimensions in both the complex and real cases, and briefly discuss the 4D quantum Hall effect. PubDate: 2016-12-01 DOI: 10.1007/s00023-016-0505-6

Abstract: Abstract We prove in a simple and coordinate-free way the equivalence between the classical definitions of the mass or of the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case. PubDate: 2016-12-01 DOI: 10.1007/s00023-016-0494-5

Abstract: Abstract In this note, we study a fractional Poisson–Nernst–Planck equation modeling a semiconductor device. We prove several decay estimates for the Lebesgue and Sobolev norms in one, two and three dimensions. We also provide the first term of the asymptotic expansion as \({t\rightarrow\infty}\) . PubDate: 2016-12-01 DOI: 10.1007/s00023-016-0493-6

Authors:Xuecheng Wang 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: 2016-11-24 DOI: 10.1007/s00023-016-0538-x

Authors:Dmitry Pelinovsky; Guido Schneider 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: 2016-11-24 DOI: 10.1007/s00023-016-0536-z

Authors:Håkan Andréasson; David Fajman; Maximilian Thaller Abstract: Abstract We prove existence of spherically symmetric, static, self-gravitating photon shells as solutions to the massless Einstein–Vlasov system. The solutions are highly relativistic in the sense that the ratio 2m(r) / r is close to 8 / 9, where m(r) is the Hawking mass and r is the area radius. In 1955 Wheeler constructed, by numerical means, so-called idealized spherically symmetric geons, i.e., solutions of the Einstein–Maxwell equations for which the energy momentum tensor is spherically symmetric on a time average. The structure of these solutions is such that the electromagnetic field is confined to a thin shell for which the ratio 2m / r is close to 8 / 9, i.e., the solutions are highly relativistic photon shells. The solutions presented in this work provide an alternative model for photon shells or idealized spherically symmetric geons. PubDate: 2016-11-22 DOI: 10.1007/s00023-016-0531-4

Authors:Jussi Behrndt; Rupert L. Frank; Christian Kühn; Vladimir Lotoreichik; Jonathan Rohleder 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: 2016-11-21 DOI: 10.1007/s00023-016-0532-3

Authors:Etera R. Livine 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: 2016-11-19 DOI: 10.1007/s00023-016-0535-0

Authors:Daniel Puzzuoli; John Watrous 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: 2016-11-17 DOI: 10.1007/s00023-016-0537-y

Authors:Marco Benini; Alexander Schenkel 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: 2016-11-17 DOI: 10.1007/s00023-016-0533-2

Authors:Michael Goldberg; William R. Green 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: 2016-11-16 DOI: 10.1007/s00023-016-0534-1

Authors:Alexandre Belin; Christoph A. Keller; Alexander Maloney Abstract: Abstract The space of permutation orbifolds is a simple landscape of two-dimensional CFTs, generalizing the well-known symmetric orbifolds. We consider constraints which a permutation orbifold with large central charge must obey in order to be holographically dual to a weakly coupled (but possibly stringy) theory of gravity in AdS. We then construct explicit examples of permutation orbifolds which obey these constraints. In our constructions, the spectrum remains finite at large N, but differs qualitatively from that of symmetric orbifolds. We also discuss under what conditions the correlation functions factorize at large N and thus reduce to those of a generalized free field in AdS. We show that this happens not just for symmetric orbifolds, but also for permutation groups which act “democratically” in a sense which we define. PubDate: 2016-11-15 DOI: 10.1007/s00023-016-0529-y