Abstract: Given a Riemannian 3-ball \(({\bar{B}}, g)\) of nonnegative scalar curvature, Bartnik conjectured that \(({\bar{B}}, g)\) admits an asymptotically flat (AF) extension (without horizons) of the least possible ADM mass and that such a mass minimizer is an AF solution to the static vacuum Einstein equations, uniquely determined by natural geometric conditions on the boundary data of \(({\bar{B}}, g)\) . We prove the validity of the second statement, i.e., such mass minimizers, if they exist, are indeed AF solutions of the static vacuum equations. On the other hand, we prove that the first statement is not true in general; there is a rather large class of bodies \(({\bar{B}}, g)\) for which a minimal mass extension does not exist. PubDate: 2019-03-20
Abstract: Consider a Dirac operator defined on the whole plane with a mass term of size m supported outside a domain \(\Omega \) . We give a simple proof for the norm resolvent convergence, as m goes to infinity, of this operator to a Dirac operator defined on \(\Omega \) with infinite-mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials. PubDate: 2019-03-19
Abstract: Schrödinger operators on metric graphs with delta couplings at the vertices are studied. We discuss which potential and which distribution of delta couplings on a given graph maximise the ground state energy, provided the integral of the potential and the sum of strengths of the delta couplings are fixed. It appears that the optimal potential if it exists is a constant function on its support formed by a set of intervals separated from the vertices. In the case where the optimal configuration does not exist explicit optimising sequences are presented. PubDate: 2019-03-18
Abstract: We prove smoothness of the correlation functions in probabilistic Liouville Conformal Field Theory. Our result is a step towards proving that the correlation functions satisfy the higher Ward identities and the higher BPZ equations, predicted by the Conformal Bootstrap approach to Conformal Field Theory. PubDate: 2019-03-16
Abstract: In this paper, we are concerned with an elliptic system arising from the Einstein–Maxwell–Higgs model which describes electromagnetic dynamics coupled with gravitational fields in spacetime. Reducing this system to a single equation and setting up the radial ansatz, we classify solutions into three cases: topological solutions, nontopological solutions of type I, and nontopological solutions of type II. There are two important constants: \(a>0\) representing the gravitational constant and \(N\ge 0\) representing the total string number. When \(0\le aN<2\) , we give a complete classification of all possible solutions and prove the uniqueness of solutions for a given decay rate. In particular, we obtain a new class of topological solitons, with nonstandard asymptotic value \(\sigma <0\) at infinity, when the total string number is sufficiently large such that \(1<aN<2\) . We also prove the multiple existence of solutions for a given decay rate in the case \(aN \ge 2\) . Our classification improves previous results which are known only for the case \(0<aN<1\) . PubDate: 2019-03-16
Abstract: In this note, we prove decay for the spin ± 1 Teukolsky equations on the Schwarzschild spacetime. These equations are those satisfied by the extreme components ( \(\alpha \) and \({\underline{\alpha }}\) ) of the Maxwell field, when expressed with respect to a null frame. The subject has already been addressed in the literature, and the interest in the present approach lies in the connection with the recent work by Dafermos, Holzegel and Rodnianski on linearized gravity (Dafermos et al. in The linear stability of the Schwarzschild solution to gravitational perturbations, 2016. arXiv:1601.06467). In analogy with the spin \(\pm 2\) case, it seems difficult to directly prove Morawetz estimates for solutions to the spin \(\pm 1\) Teukolsky equations. By performing a differential transformation on the extreme components \(\alpha \) and \({\underline{\alpha }}\) , we obtain quantities which satisfy a Fackerell–Ipser Equation, which does admit a straightforward Morawetz estimate and is the key to the decay estimates. This approach is exactly analogous to the strategy appearing in the aforementioned work on linearized gravity. We achieve inverse polynomial decay estimates by a streamlined version of the physical space \(r^p\) method of Dafermos and Rodnianski. Furthermore, we are also able to prove decay for all the components of the Maxwell system. The transformation that we use is a physical space version of a fixed-frequency transformation which appeared in the work of Chandrasekhar (Proc R Soc Lond Ser A 348(1652):39–55, 1976). The present note is a version of the author’s master thesis and also serves the “pedagogical” purpose to be as complete as possible in the presentation. PubDate: 2019-03-11
Abstract: For soliton cellular automata, we give a uniform description and proofs of the solitons, the scattering rule of two solitons, and the phase shift using rigged configurations in a number of special cases. In particular, we prove these properties for the soliton cellular automata using \(B^{r,1}\) when r is adjacent to 0 in the Dynkin diagram or there is a Dynkin diagram automorphism sending r to 0. PubDate: 2019-03-11
Abstract: We present Pieri rules for the Jack polynomials in superspace. The coefficients in the Pieri rules are, except for an extra determinant, products of quotients of linear factors in \(\alpha \) (expressed, as in the usual Jack polynomial case, in terms of certain hook lengths in a Ferrers’ diagram). We show that, surprisingly, the extra determinant is related to the partition function of the 6-vertex model. We give, as a conjecture, the Pieri rules for the Macdonald polynomials in superspace. PubDate: 2019-03-07
Abstract: Integrable many-body systems of Ruijsenaars–Schneider–van Diejen type displaying action-angle duality are derived by Hamiltonian reduction of the Heisenberg double of the Poisson–Lie group \(\mathrm{SU}(2n)\) . New global models of the reduced phase space are described, revealing non-trivial features of the two systems in duality with one another. For example, after establishing that the symplectic vector space \(\mathbb {C}^n\simeq \mathbb {R}^{2n}\) underlies both global models, it is seen that for both systems the action variables generate the standard torus action on \(\mathbb {C}^n\) , and the fixed point of this action corresponds to the unique equilibrium positions of the pertinent systems. The systems in duality are found to be non-degenerate in the sense that the functional dimension of the Poisson algebra of their conserved quantities is equal to half the dimension of the phase space. The dual of the deformed Sutherland system is shown to be a limiting case of a van Diejen system. PubDate: 2019-03-06
Abstract: Static spherically symmetric solutions to the Einstein–Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the latter case, the matter extends globally with the density approaching zero at infinity. The asymptotic behavior largely depends on the equation of state of the fluid and is still poorly understood. While a few such unbounded solutions are known to be asymptotically flat with finite ADM mass, the vast majority are not. We provide a full geometric description of the asymptotic behavior of static spherically symmetric perfect fluid solutions with linear equations of state and polytropic-type equations of state with index \(n>5\) . In order to capture the asymptotic behavior, we introduce a notion of scaled quasi-asymptotic flatness, which encompasses the notion of asymptotic conicality. In particular, these spacetimes are asymptotically simple. PubDate: 2019-03-01
Abstract: We consider the dynamics of N interacting bosons initially forming a Bose–Einstein condensate. Due to an external trapping potential, the bosons are strongly confined in two dimensions, where the transverse extension of the trap is of order \(\varepsilon \) . The non-negative interaction potential is scaled such that its range and its scattering length are both of order \((N/\varepsilon ^2)^{-1}\) , corresponding to the Gross–Pitaevskii scaling of a dilute Bose gas. We show that in the simultaneous limit \(N\rightarrow \infty \) and \(\varepsilon \rightarrow 0\) , the dynamics preserve condensation and the time evolution is asymptotically described by a Gross–Pitaevskii equation in one dimension. The strength of the nonlinearity is given by the scattering length of the unscaled interaction, multiplied with a factor depending on the shape of the confining potential. For our analysis, we adapt a method by Pickl (Rev Math Phys 27(01):1550003, 2015) to the problem with dimensional reduction and rely on the derivation of the one-dimensional NLS equation for interactions with softer scaling behaviour in Boßmann (Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons. arXiv preprint, 2018. arXiv:1803.11011). PubDate: 2019-03-01
Abstract: The Strong Cosmic Censorship conjecture states that for generic initial data to Einstein’s field equations, the maximal globally hyperbolic development is inextendible. We prove this conjecture in the class of orthogonal Bianchi class B perfect fluids and vacuum spacetimes, by showing that unboundedness of certain curvature invariants such as the Kretschmann scalar is a generic property. The only spacetimes where this scalar remains bounded exhibit local rotational symmetry or are of plane wave equilibrium type. We further investigate the qualitative behaviour of solutions towards the initial singularity. To this end, we work in the expansion-normalised variables introduced by Hewitt–Wainwright and show that a set of full measure, which is also a countable intersection of open and dense sets in the state space, yields convergence to a specific subarc of the Kasner parabola. We further give an explicit construction enabling the translation between these variables and geometric initial data to Einstein’s equations. PubDate: 2019-03-01
Abstract: In general relativity, there have been a number of successful constructions for asymptotically flat metrics with a certain background foliation. In particular, Lin (Calc Var 49(3–4):1309–1335, 2014) used a foliation by the Ricci flow on 2-spheres to establish an asymptotically flat extension, and Sormani and Lin (Ann Henri Poincaré 17(10):2783–2800, 2016) proved useful results with this extension. In this paper, we construct asymptotically hyperbolic 3-metrics with the Ricci flow foliation inspired by Lin’s work. We also study the rigid case when the Hawking mass of the inner surface of the manifold agrees with its total mass as in Sormani and Lin (2016). PubDate: 2019-03-01
Abstract: In this article, we estimate the quasi-local energy with reference to the Minkowski spacetime (Wang and Yau in Phys Rev Lett 102(2):021101, 2009; Commun Math Phys 288(3):919–942, 2009), the anti-de Sitter spacetime (Chen et al. in Commun Anal Geom, 2016. arXiv:1603.02975), or the Schwarzschild spacetime (Chen et al. in Adv Theor Math Phys 22(1):1–23, 2018). In each case, the reference spacetime admits a conformal Killing–Yano 2-form which facilitates the application of the Minkowski formula in Wang et al. (J Differ Geom 105(2):249–290, 2017) to estimate the quasi-local energy. As a consequence of the positive mass theorems in Liu and Yau (J Am Math Soc 19(1):181–204, 2006) and Shi and Tam (Class Quantum Gravity 24(9):2357–2366, 2007) and the above estimate, we obtain rigidity theorems which characterize the Minkowski spacetime and the hyperbolic space. PubDate: 2019-03-01
Abstract: This work presents some results about Wick polynomials of a vector field renormalization in locally covariant algebraic quantum field theory in curved spacetime. General vector fields are pictured as sections of natural vector bundles over globally hyperbolic spacetimes and quantized through the known functorial machinery in terms of local \(*\) -algebras. These quantized fields may be defined on spacetimes with given classical background fields, also sections of natural vector bundles, in addition to the Lorentzian metric. The mass and the coupling constants are in particular viewed as background fields. Wick powers of the quantized vector field are axiomatically defined imposing in particular local covariance, scaling properties, and smooth dependence on smooth perturbation of the background fields. A general classification theorem is established for finite renormalization terms (or counterterms) arising when comparing different solutions satisfying the defining axioms of Wick powers. The result is specialized to the case of general tensor fields. In particular, the case of a vector Klein–Gordon field and the case of a scalar field renormalized together with its derivatives are discussed as examples. In each case, a more precise statement about the structure of the counterterms is proved. The finite renormalization terms turn out to be finite-order polynomials tensorially and locally constructed with the backgrounds fields and their covariant derivatives whose coefficients are locally smooth functions of polynomial scalar invariants constructed from the so-called marginal subset of the background fields. The notion of local smooth dependence on polynomial scalar invariants is made precise in the text. Our main technical tools are based on the Peetre–Slovák theorem characterizing differential operators and on the classification of smooth invariants on representations of reductive Lie groups. PubDate: 2019-03-01
Abstract: We consider the perturbative construction, proposed in Fredenhagen and Lindner (Commun Math Phys 332:895, 2014), for a thermal state \(\Omega _{\beta ,\lambda V\{f\}}\) for the theory of a real scalar Klein–Gordon field \(\phi \) with interacting potential \(V\{f\}\) . Here, f is a space-time cut-off of the interaction V, and \(\lambda \) is a perturbative parameter. We assume that V is quadratic in the field \(\phi \) and we compute the adiabatic limit \(f\rightarrow 1\) of the state \(\Omega _{\beta ,\lambda V\{f\}}\) . The limit is shown to exist; moreover, the perturbative series in \(\lambda \) sums up to the thermal state for the corresponding (free) theory with potential V. In addition, we exploit the same methods to address a similar computation for the non-equilibrium steady state (NESS) Ruelle (J Stat Phys 98:57–75, 2000) recently constructed in Drago et al. (Commun Math Phys 357:267, 2018). PubDate: 2019-03-01
Abstract: We show that there is generically non-uniqueness for the anisotropic Calderón problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show that given a smooth compact connected Riemannian manifold with boundary (M, g) of dimension \(n\ge 3\) , there exist in the conformal class of g an infinite number of Riemannian metrics \(\tilde{g}\) such that their corresponding DN maps at a fixed frequency coincide when the Dirichlet data \(\Gamma _D\) and Neumann data \(\Gamma _N\) are measured on disjoint sets and satisfy \(\overline{\Gamma _D \cup \Gamma _N} \ne \partial M\) . The conformal factors that lead to these non-uniqueness results for the anisotropic Calderón problem satisfy a nonlinear elliptic PDE of Yamabe type on the original manifold (M, g) and are associated with a natural but subtle gauge invariance of the anisotropic Calderón problem with data on disjoint sets. We then construct a large class of counterexamples to uniqueness in dimension \(n\ge 3\) to the anisotropic Calderón problem at fixed frequency with data on disjoint sets and modulo this gauge invariance. This class consists in cylindrical Riemannian manifolds with boundary having two ends (meaning that the boundary has two connected components), equipped with a suitably chosen warped product metric. PubDate: 2019-03-01
Abstract: For many completely positive maps repeated compositions will eventually become entanglement breaking. To quantify this behaviour we develop a technique based on the Schmidt number: If a completely positive map breaks the entanglement with respect to any qubit ancilla, then applying it to part of a bipartite quantum state will result in a Schmidt number bounded away from the maximum possible value. Iterating this result puts a successively decreasing upper bound on the Schmidt number arising in this way from compositions of such a map. By applying this technique to completely positive maps in dimension three that are also completely copositive we prove the so-called PPT squared conjecture in this dimension. We then give more examples of completely positive maps where our technique can be applied, e.g. maps close to the completely depolarizing map, and maps of low rank. Finally, we study the PPT squared conjecture in more detail, establishing equivalent conjectures related to other parts of quantum information theory, and we prove the conjecture for Gaussian quantum channels. PubDate: 2019-02-14
Abstract: We develop a scattering theory for the linear wave equation \(\Box _g \psi = 0 \) on the interior of Reissner–Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution \(\psi \) on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resulting radial o.d.e. over all frequencies \(\omega \) and \(\ell \) . This is non-trivial because the natural T conservation law is sign-indefinite in the black hole interior. In the physical space picture, our results imply that the Cauchy evolution from the event horizon to the Cauchy horizon is a Hilbert space isomorphism, where the past (resp. future) Hilbert space is defined by the finiteness of the degenerate T energy fluxes on both components of the event (resp. Cauchy) horizon. Finally, we prove that, in contrast to the above, for a generic set of cosmological constants \(\Lambda \) , there is no analogous finite T energy scattering theory for either the linear wave equation or the Klein–Gordon equation with conformal mass on the (anti-) de Sitter–Reissner–Nordström interior. PubDate: 2019-02-13
Abstract: Motivated by the recent contribution (Bauer and Bernard in Annales Henri Poincaré 19:653–693, 2018), we study the scaling limit behavior of a class of one-dimensional stochastic differential equations which has a unique attracting point subject to a small additional repulsive perturbation. Problems of this type appear in the analysis of continuously monitored quantum systems. We extend the results of Bauer and Bernard (Annales Henri Poincaré 19:653–693, 2018) and prove a general result concerning the convergence to a homogeneous Poisson process using only classical probabilistic tools. PubDate: 2019-02-11
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