Authors:Rupert L. Frank; Marius Lemm Pages: 2285 - 2340 Abstract: Abstract This paper consists of three parts. In part I, we microscopically derive Ginzburg–Landau (GL) theory from BCS theory for translation-invariant systems in which multiple types of superconductivity may coexist. Our motivation are unconventional superconductors. We allow the ground state of the effective gap operator \({K_{T_c}+V}\) to be n-fold degenerate and the resulting GL theory then couples n order parameters. In part II, we study examples of multi-component GL theories which arise from an isotropic BCS theory. We study the cases of (a) pure d-wave order parameters and (b) mixed (s + d)-wave order parameters, in two and three-dimensions. In part III, we present explicit choices of spherically symmetric interactions V which produce the examples in part II. In fact, we find interactions V which produce ground state sectors of \({K_{T_c}+V}\) of arbitrary angular momentum, for open sets of of parameter values. This is in stark contrast with Schrödinger operators \({-\nabla^2+V}\) , for which the ground state is always non-degenerate. Along the way, we prove the following fact about Bessel functions: At its first maximum, a half-integer Bessel function is strictly larger than all other half-integer Bessel functions. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0473-x Issue No:Vol. 17, No. 9 (2016)

Authors:Ram Band; David Fajman Pages: 2379 - 2407 Abstract: Abstract A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition. An alternative partition is revealed by considering a set of distinguished gradient flow lines of the eigenfunction—those which are connected to saddle points. These give rise to Neumann domains. We establish complementary definitions for Neumann domains and Neumann lines and use basic Morse homology to prove their fundamental topological properties. We study the eigenfunction restrictions to these domains. Their zero set, critical points and spectral properties allow to discuss some aspects of counting the number of Neumann domains and estimating their geometry. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0468-7 Issue No:Vol. 17, No. 9 (2016)

Authors:Tadayoshi Adachi; Masaki Kawamoto Pages: 2409 - 2438 Abstract: Abstract In this paper, we study the quantum dynamics of a charged particle in the plane in the presence of a periodically pulsed magnetic field perpendicular to the plane. We show that by controlling the cycle when the magnetic field is switched on and off appropriately, the result of the asymptotic completeness of wave operators can be obtained under the assumption that the potential V satisfies the decaying condition \({ V(x) \le C(1 + x )^{-\rho}}\) for some \({\rho > 0}\) . PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0457-x Issue No:Vol. 17, No. 9 (2016)

Authors:James B. Kennedy; Pavel Kurasov; Gabriela Malenová; Delio Mugnolo Pages: 2439 - 2473 Abstract: Abstract We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0460-2 Issue No:Vol. 17, No. 9 (2016)

Authors:Yan-Hui Qu Pages: 2475 - 2511 Abstract: Abstract We study the spectral properties of the Sturm Hamiltolian of eventually constant type, which includes the Fibonacci Hamiltonian. Let s be the Hausdorff dimension of the spectrum. For V > 20, we show that the restriction of the s-dimensional Hausdorff measure to the spectrum is a Gibbs type measure; the density of states measure is a Markov measure. Based on the fine structures of these measures, we show that both measures are exact dimensional; we obtain exact asymptotic behaviors for the optimal Hölder exponent and the Hausdorff dimension of the density of states measure and for the Hausdorff dimension of the spectrum. As a consequence, if the frequency is not silver number type, then for V big enough, we establish strict inequalities between these three spectral characteristics. We achieve them by introducing an auxiliary symbolic dynamical system and applying the thermodynamical and multifractal formalisms of almost additive potentials. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0462-0 Issue No:Vol. 17, No. 9 (2016)

Authors:Joachim Asch; Olivier Bourget; Victor Cortés; Claudio Fernandez Pages: 2513 - 2527 Abstract: Abstract One manifestation of quantum resonances is a large sojourn time, or autocorrelation, for states which are initially localized. We elaborate on Lavine’s time-energy uncertainty principle and give an estimate on the sojourn time. For the case of perturbed embedded eigenstates the bound is explicit and involves Fermi’s Golden Rule. It is valid for a very general class of systems. We illustrate the theory by applications to resonances for time-dependent systems including the AC Stark effect as well as multistate systems. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0467-8 Issue No:Vol. 17, No. 9 (2016)

Authors:Mohamed Lazhar Tayeb Pages: 2529 - 2553 Abstract: Abstract The approximation by diffusion and homogenization of the initial-boundary value problem of the Vlasov–Poisson–Fokker–Planck model is studied for a given velocity field with spatial macroscopic and microscopic variations. The L1-contraction property of the Fokker–Planck operator and a two-scale Hybrid-Hilbert expansion are used to prove the convergence towards a homogenized Drift–Diffusion equation and to exhibit a rate of convergence. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0484-7 Issue No:Vol. 17, No. 9 (2016)

Authors:Varga Kalantarov; Anton Savostianov; Sergey Zelik Pages: 2555 - 2584 Abstract: Abstract The dissipative wave equation with a critical quintic non-linearity in smooth bounded three-dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0480-y Issue No:Vol. 17, No. 9 (2016)

Authors:Johannes Kautzsch; Marc Kesseböhmer; Tony Samuel Pages: 2585 - 2621 Abstract: Abstract We consider a family \({\{T_{r}: [0, 1] \circlearrowleft \}_{r\in[0, 1]}}\) of Markov interval maps interpolating between the tent map \({T_{0}}\) and the Farey map \({T_{1}}\) . Letting \({\mathcal{P}_{r}}\) denote the Perron–Frobenius operator of \({T_{r}}\) , we show, for \({\beta \in [0, 1]}\) and \({\alpha \in (0, 1)}\) , that the asymptotic behaviour of the iterates of \({\mathcal{P}_{r}}\) applied to observables with a singularity at \({\beta}\) of order \({\alpha}\) is dependent on the structure of the \({\omega}\) -limit set of \({\beta}\) with respect to \({T_{r}}\) . The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities. PubDate: 2016-09-01 DOI: 10.1007/s00023-015-0451-8 Issue No:Vol. 17, No. 9 (2016)

Authors:Nathan Benjamin; Miranda C. N. Cheng; Shamit Kachru; Gregory W. Moore; Natalie M. Paquette Pages: 2623 - 2662 Abstract: Abstract We describe general constraints on the elliptic genus of a 2d supersymmetric conformal field theory which has a gravity dual with large radius in Planck units. We give examples of theories which do and do not satisfy the bounds we derive, by describing the elliptic genera of symmetric product orbifolds of K3, product manifolds, certain simple families of Calabi–Yau hypersurfaces, and symmetric products of the “Monster CFT”. We discuss the distinction between theories with supergravity duals and those whose duals have strings at the scale set by the AdS curvature. Under natural assumptions, we attempt to quantify the fraction of (2,2) supersymmetric conformal theories which admit a weakly curved gravity description, at large central charge. PubDate: 2016-10-01 DOI: 10.1007/s00023-016-0469-6 Issue No:Vol. 17, No. 10 (2016)

Authors:Nathan Benjamin; Sarah M. Harrison; Shamit Kachru; Natalie M. Paquette; Daniel Whalen Pages: 2663 - 2697 Abstract: Abstract Superstring compactification on a manifold of Spin(7) holonomy gives rise to a 2d worldsheet conformal field theory with an extended supersymmetry algebra. The \({\mathcal{N} = 1}\) superconformal algebra is extended by additional generators of spins 2 and 5/2, and instead of just superconformal symmetry one has a c = 12 realization of the symmetry group \({\mathcal{S}W(3/2,2)}\) . In this paper, we compute the characters of this supergroup and decompose the elliptic genus of a general Spin(7) compactification in terms of these characters. We find suggestive relations to various sporadic groups, which are made more precise in a companion paper. PubDate: 2016-10-01 DOI: 10.1007/s00023-015-0454-5 Issue No:Vol. 17, No. 10 (2016)

Authors:Giovanni Morchio; Franco Strocchi Pages: 2699 - 2739 Abstract: Abstract The scattering of photons and heavy classical Coulomb interacting particles, with realistic particle–photon interaction (without particle recoil) is studied adopting the Koopman formulation for the particles. The model is translation invariant and allows for a complete control of the Dollard strategy devised by Kulish–Faddeev and Rohrlich (KFR) for QED: in the adiabatic formulation, the Møller operators exist as strong limits and interpolate between the dynamics and a non-free asymptotic dynamics, which is a unitary group; the S-matrix is non-trivial and exhibits the factorization of all the infrared divergences. The implications of the KFR strategy on the open questions of the LSZ asymptotic limits in QED are derived in the field theory version of the model, with the charged particles described by second quantized fields: i) asymptotic limits of the charged fields, \({\Psi_{{\rm out}/{\rm in}}(x)}\) , are obtained as strong limits of modified LSZ formulas, with corrections given by a Coulomb phase operator and an exponential of the photon field; ii) free asymptotic electromagnetic fields, \({B_{{\rm out}/{\rm in}}(x)}\) , are given by the massless LSZ formula, as in Buchholz approach; iii) the asymptotic field algebras are a semidirect product of the canonical algebras generated by \({B_{{\rm out}/{\rm in}}}\) , \({\Psi_{{\rm out}/{\rm in}}}\) ; iv) on the asymptotic spaces, the Hamiltonian is the sum of the free (commuting) Hamiltonians of \({B_{{\rm out}/{\rm in}}}\) , \({\Psi_{{\rm out}/{\rm in}}}\) and the same holds for the generators of the space translations. PubDate: 2016-10-01 DOI: 10.1007/s00023-016-0486-5 Issue No:Vol. 17, No. 10 (2016)

Authors:Rinat Kashaev; Marcos Mariño; Szabolcs Zakany Pages: 2741 - 2781 Abstract: Abstract The quantization of mirror curves to toric Calabi–Yau threefolds leads to trace class operators, and it has been conjectured that the spectral properties of these operators provide a non-perturbative realization of topological string theory on these backgrounds. In this paper, we find an explicit form for the integral kernel of the trace class operator in the case of local \({{\mathbb{P}}^1 \times {\mathbb{P}}^1}\) , in terms of Faddeev’s quantum dilogarithm. The matrix model associated to this integral kernel is an \({O(2)}\) model, which generalizes the ABJ(M) matrix model. We find its exact planar limit, and we provide detailed evidence that its \({1/N}\) expansion captures the all genus topological string free energy on local \({{\mathbb{P}}^1 \times {\mathbb{P}}^1}\) . PubDate: 2016-10-01 DOI: 10.1007/s00023-016-0471-z Issue No:Vol. 17, No. 10 (2016)

Authors:Chen-Yun Lin; Christina Sormani Pages: 2783 - 2800 Abstract: Abstract We provide estimates on the Bartnik mass of constant mean curvature surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new notion we call the asphericity mass. The asphericity mass is defined by applying Hamilton’s modified Ricci flow and depends only upon the restricted metric of the surface and not on its mean curvature. The theorem is proven by studying a class of asymptotically flat Riemannian manifolds foliated by surfaces satisfying Hamilton’s modified Ricci flow with prescribed scalar curvature. Such manifolds were first constructed by the first author in her dissertation conducted under the supervision of M. T. Wang. We make a further study of this class of manifolds which we denote Ham3, bounding the ADM masses of such manifolds and analyzing the rigid case when the Hawking mass of the inner surface of the manifold agrees with its ADM mass. PubDate: 2016-10-01 DOI: 10.1007/s00023-016-0483-8 Issue No:Vol. 17, No. 10 (2016)

Authors:Piotr T. Chruściel; James D. E. Grant; Ettore Minguzzi Pages: 2801 - 2824 Abstract: Abstract We show differentiability of a class of Geroch’s volume functions on globally hyperbolic manifolds. Furthermore, we prove that every volume function satisfies a local anti-Lipschitz condition over causal curves, and that locally Lipschitz time functions which are locally anti-Lipschitz can be uniformly approximated by smooth time functions with timelike gradient. Finally, we prove that in stably causal space-times Hawking’s time function can be uniformly approximated by smooth time functions with timelike gradient. PubDate: 2016-10-01 DOI: 10.1007/s00023-015-0448-3 Issue No:Vol. 17, No. 10 (2016)

Authors:Alessandro Carlotto Pages: 2825 - 2847 Abstract: Abstract In this article, we investigate the restrictions imposed by the dominant energy condition (DEC) on the topology and conformal type of possibly non-compact marginally outer trapped surfaces (thus extending Hawking’s classical theorem on the topology of black holes). We first prove that an unbounded, stable marginally outer trapped surface in an initial data set (M, g, k) obeying the dominant energy condition is conformally diffeomorphic to either the plane \({\mathbb{C}}\) or to the cylinder \({\mathbb{A}}\) and in the latter case infinitesimal rigidity holds. As a corollary, when the DEC holds strictly, this rules out the existence of trapped regions with cylindrical boundary. In the second part of the article, we restrict our attention to asymptotically flat data (M, g, k) and show that, in that setting, the existence of an unbounded, stable marginally outer trapped surface essentially never occurs unless in a very specific case, since it would force an isometric embedding of (M, g, k) into the Minkowski spacetime as a space-like slice. PubDate: 2016-10-01 DOI: 10.1007/s00023-016-0477-6 Issue No:Vol. 17, No. 10 (2016)

Authors:Thierry Daudé; François Nicoleau Pages: 2849 - 2904 Abstract: Abstract We study inverse scattering problems at a fixed energy for radial Schrödinger operators on \({\mathbb{R}^n}\) , \({n \geq 2}\) . First, we consider the class \({\mathcal{A}}\) of potentials q(r) which can be extended analytically in \({\Re z \geq 0}\) such that \({\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}}\) , \({\rho > \frac{3}{2}}\) . If q and \({\tilde{q}}\) are two such potentials and if the corresponding phase shifts \({\delta_l}\) and \({\tilde{\delta}_l}\) are super-exponentially close, then \({q=\tilde{q}}\) . Second, we study the class of potentials q(r) which can be split into q(r) = q 1(r) + q 2(r) such that q 1(r) has compact support and \({q_2 (r) \in \mathcal{A}}\) . If q and \({\tilde{q}}\) are two such potentials, we show that for any fixed \({a>0, {\delta_l - \tilde{\delta}_l \ = \ o \left(\frac{1}{l^{n-3}}\ \left({\frac{ae}{2l}}\right)^{2l}\right)}}\) when \({l \rightarrow +\infty}\) if and only if \({q(r)=\tilde{q}(r)}\) for almost all \({r \geq a}\) . The proofs are close in spirit with the celebrated Borg–Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in \({\Re z \geq 0}\) with \({\mid q(z)\mid \leq C (1+ \mid z \mid)^{-\rho}}\) , \({\rho >1}\) , we show that the Regge poles are confined in a vertical strip in the complex plane. PubDate: 2016-10-01 DOI: 10.1007/s00023-015-0453-6 Issue No:Vol. 17, No. 10 (2016)

Authors:Miguel Ballesteros; Ricardo Weder Pages: 2905 - 2950 Abstract: Abstract We analyze spin-0 relativistic scattering of charged particles propagating in the exterior, \({\Lambda \subset \mathbb{R}^3}\) , of a compact obstacle \({K \subset \mathbb{R}^3}\) . The connected components of the obstacle are handlebodies. The particles interact with an electromagnetic field in Λ and an inaccessible magnetic field localized in the interior of the obstacle (through the Aharonov–Bohm effect). We obtain high-momenta estimates, with error bounds, for the scattering operator that we use to recover physical information: we give a reconstruction method for the electric potential and the exterior magnetic field and prove that, if the electric potential vanishes, circulations of the magnetic potential around handles (or equivalently, by Stokes’ theorem, magnetic fluxes over transverse sections of handles) of the obstacle can be recovered, modulo 2π. We additionally give a simple formula for the high momenta limit of the scattering operator in terms of certain magnetic fluxes, in the absence of electric potential. If the electric potential does not vanish, the magnetic fluxes on the handles above referred can be only recovered modulo π and the simple expression of the high-momenta limit of the scattering operator does not hold true. PubDate: 2016-10-01 DOI: 10.1007/s00023-016-0466-9 Issue No:Vol. 17, No. 10 (2016)

Authors:Daniel M. Elton Pages: 2951 - 2973 Abstract: Abstract We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl–Dirac operators on \({\mathbb{R}^3}\) . In particular, we are interested in those operators \({\mathcal{D}_B}\) for which the associated magnetic field \({B}\) is given by pulling back a two-form \({\beta}\) from the sphere \({\mathbb{S}^2}\) to \({\mathbb{R}^3}\) using a combination of the Hopf fibration and inverse stereographic projection. If \({\int_{\mathbb{s}^2} \beta \neq 0}\) , we show that $$\sum_{0 \leq t \leq T} {\rm dim Ker} \mathcal{D}{tB}=\frac{T^2}{8\pi^2}\,\Big \int_{\mathbb{S}^2}\beta\Big \,\int_{\mathbb{S}^2} {\beta} +o(T^2)$$ as \({T\to+\infty}\) . The result relies on Erdős and Solovej’s characterisation of the spectrum of \({\mathcal{D}_{tB}}\) in terms of a family of Dirac operators on \({\mathbb{S}^2}\) , together with information about the strong field localisation of the Aharonov–Casher zero modes of the latter. PubDate: 2016-10-01 DOI: 10.1007/s00023-016-0478-5 Issue No:Vol. 17, No. 10 (2016)

Authors:Daniela Cadamuro; Yoh Tanimoto Abstract: Abstract We construct candidates for observables in wedge-shaped regions for a class of 1 + 1-dimensional integrable quantum field theories with bound states whose S-matrix is diagonal, by extending our previous methods for scalar S-matrices. Examples include the Z(N)-Ising models, the A N−1-affine Toda field theories and some S-matrices with CDD factors. We show that these candidate operators which are associated with elementary particles commute weakly on a dense domain. For the models with two species of particles, we can take a larger domain of weak commutativity and give an argument for the Reeh–Schlieder property. PubDate: 2016-09-09 DOI: 10.1007/s00023-016-0515-4