Authors:Vitor Araújo; Ian Melbourne Pages: 2975 - 3004 Abstract: Abstract We prove exponential decay of correlations for a class of \({C^{1+\alpha}}\) uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing. PubDate: 2016-11-01 DOI: 10.1007/s00023-016-0482-9 Issue No:Vol. 17, No. 11 (2016)

Authors:Andrey Gogolev; Jean-François Lafont Pages: 3005 - 3026 Abstract: Abstract We show that the product of infranilmanifolds with certain aspherical closed manifolds do not support Anosov diffeomorphisms. As a special case, we obtain that products of a nilmanifold and negatively curved manifolds of dimension at least 3 do not support Anosov diffeomorphisms. PubDate: 2016-11-01 DOI: 10.1007/s00023-016-0492-7 Issue No:Vol. 17, No. 11 (2016)

Authors:Jeremiah Buckley; Igor Wigman Pages: 3027 - 3062 Abstract: Abstract We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov–Sodin’s results for random fields and Bourgain’s de-randomisation procedure we establish a precise asymptotic result for “generic” eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions. PubDate: 2016-11-01 DOI: 10.1007/s00023-016-0476-7 Issue No:Vol. 17, No. 11 (2016)

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)

Authors:Semyon Dyatlov; Colin Guillarmou Pages: 3089 - 3146 Abstract: Abstract We define Pollicott–Ruelle resonances for geodesic flows on noncompact asymptotically hyperbolic negatively curved manifolds, as well as for more general open hyperbolic systems related to Axiom A flows. These resonances are the poles of the meromorphic continuation of the resolvent of the generator of the flow and they describe decay of classical correlations. As an application, we show that the Ruelle zeta function extends meromorphically to the entire complex plane. PubDate: 2016-11-01 DOI: 10.1007/s00023-016-0491-8 Issue No:Vol. 17, No. 11 (2016)

Authors:Aleksey Kostenko; Gerald Teschl; Julio H. Toloza Pages: 3147 - 3176 Abstract: Abstract We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum. PubDate: 2016-11-01 DOI: 10.1007/s00023-016-0474-9 Issue No:Vol. 17, No. 11 (2016)

Authors:Alba Grassi; Yasuyuki Hatsuda; Marcos Mariño Pages: 3177 - 3235 Abstract: Abstract We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi–Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov–Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local \({{\mathbb{P}}^2}\) , local \({{\mathbb{P}}^1 \times {\mathbb{P}}^1}\) and local \({{\mathbb{F}}_1}\) . In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi–Yau manifolds, in which the genus expansion emerges as a ’t Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background-independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry. PubDate: 2016-11-01 DOI: 10.1007/s00023-016-0479-4 Issue No:Vol. 17, No. 11 (2016)

Authors:Axel de Goursac; Andrea Sportiello; Adrian Tanasa Pages: 3237 - 3254 Abstract: Abstract The Jacobian Conjecture states that any locally invertible polynomial system in \({{\mathbb{C}}^n}\) is globally invertible with polynomial inverse. Bass et al. (Bull Am Math Soc 7(2):287–330, 1982) proved a reduction theorem stating that the conjecture is true for any degree of the polynomial system if it is true in degree three. This degree reduction is obtained with the price of increasing the dimension \({n}\) . We prove here a theorem concerning partial elimination of variables, which implies a reduction of the generic case to the quadratic one. The price to pay is the introduction of a supplementary parameter \({0 \leq n' \leq n}\) , parameter which represents the dimension of a linear subspace where some particular conditions on the system must hold. We first give a purely algebraic proof of this reduction result and we then expose a distinct proof, in a Quantum Field Theoretical formulation, using the intermediate field method. PubDate: 2016-11-01 DOI: 10.1007/s00023-016-0490-9 Issue No:Vol. 17, No. 11 (2016)

Authors:Koen van den Dungen; Adam Rennie Pages: 3255 - 3286 Abstract: Abstract We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reversible. We present three examples of our framework: the Dirac operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an indefinite metric); the harmonic oscillator; and the construction via the Kasparov product of an indefinite spectral triple from a family of spectral triples. This last construction corresponds to a foliation of a globally hyperbolic spacetime by spacelike hypersurfaces. PubDate: 2016-11-01 DOI: 10.1007/s00023-016-0463-z Issue No:Vol. 17, No. 11 (2016)

Authors:Julien Guillod; Peter Wittwer Pages: 3287 - 3319 Abstract: Abstract We consider the stationary incompressible Navier–Stokes equation in the half-plane with inhomogeneous boundary condition. We prove the existence of strong solutions for boundary data close to any Jeffery–Hamel solution with small flux evaluated on the boundary. The perturbation of the Jeffery–Hamel solution on the boundary has to satisfy a nonlinear compatibility condition which corresponds to the integral of the velocity field on the boundary. The first component of this integral is the flux which is an invariant quantity, but the second, called the asymmetry, is not invariant, which leads to one compatibility condition. Finally, we prove the existence of weak solutions, as well as weak–strong uniqueness for small data and provide numerical simulations. PubDate: 2016-11-01 DOI: 10.1007/s00023-016-0470-0 Issue No:Vol. 17, No. 11 (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: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)

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: 2016-10-21

Abstract: Abstract Recently, a correspondence has been proposed between spectral theory and topological strings on toric Calabi–Yau manifolds. In this paper, we develop in detail this correspondence for mirror curves of higher genus, which display many new features as compared to the genus one case studied so far. Given a curve of genus g, our quantization scheme leads to g different trace class operators. Their spectral properties are encoded in a generalized spectral determinant, which is an entire function on the Calabi–Yau moduli space. We conjecture an exact expression for this spectral determinant in terms of the standard and refined topological string amplitudes. This conjecture provides a non-perturbative definition of the topological string on these geometries, in which the genus expansion emerges in a suitable ’t Hooft limit of the spectral traces of the operators. In contrast to what happens in quantum integrable systems, our quantization scheme leads to a single quantization condition, which is elegantly encoded by the vanishing of a quantum-deformed theta function on the mirror curve. We illustrate our general theory by analyzing in detail the resolved \({\mathbb C}^3/{\mathbb Z}_5\) orbifold, which is the simplest toric Calabi–Yau manifold with a genus two mirror curve. By applying our conjecture to this example, we find new quantization conditions for quantum mechanical operators, in terms of genus two theta functions, as well as new number-theoretic properties for the periods of this Calabi–Yau. PubDate: 2016-10-21

Abstract: Abstract We introduce and study a Markov field on the edges of a graph \(\mathcal {G}\) in dimension \(d\ge 2\) whose configurations are spin networks. The field arises naturally as the edge-occupation field of a Poissonian model (a soup) of non-backtracking loops and walks characterized by a spatial Markov property such that, conditionally on the value of the edge-occupation field on a boundary set that splits the graph into two parts, the distributions of the loops and arcs contained in the two parts are independent of each other. The field has a Gibbs distribution with a Hamiltonian given by a sum of terms which involve only edges incident on the same vertex. Its free energy density and other quantities can be computed exactly, and their critical behavior analyzed, in any dimension. PubDate: 2016-10-20

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: 2016-10-20

Abstract: Abstract We improve the Lieb–Robinson bound for a wide class of quantum many-body systems with long-range interactions decaying by power law. As an application, we show that the group velocity of information propagation grows by power law in time for such systems, whereas systems with short-range interactions exhibit a finite group velocity as shown by Lieb and Robinson. PubDate: 2016-10-20

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: 2016-10-19

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: 2016-10-19

Abstract: Abstract We consider a non-self-adjoint h-differential model operator \(P_h\) in the semiclassical limit ( \(h\rightarrow 0\) ) subject to small random perturbations. Furthermore, we let the coupling constant \(\delta \) be \(\mathrm {e}^{-\frac{1}{Ch}}\le \delta \ll h^{\kappa }\) for constants \(C,\kappa >0\) suitably large. Let \(\Sigma \) be the closure of the range of the principal symbol. Previous results on the same model by Hager, Bordeaux-Montrieux and Sjöstrand show that if \(\delta \gg \mathrm {e}^{-\frac{1}{Ch}}\) there is, with a probability close to 1, a Weyl law for the eigenvalues in the interior of the of the pseudospectrum up to a distance \(\gg \left( -h\ln {\delta h}\right) ^{\frac{2}{3}}\) to the boundary of \(\Sigma \) . We study the intensity measure of the random point process of eigenvalues and prove an h-asymptotic formula for the average density of eigenvalues. With this we show that there are three distinct regions of different spectral behavior in \(\Sigma \) : the interior of the pseudospectrum is solely governed by a Weyl law, close to its boundary there is a strong spectral accumulation given by a tunneling effect followed by a region where the density decays rapidly. PubDate: 2016-10-19