Authors:Hamid Hezari; Zhiqin Lu; Julie Rowlett Pages: 3759 - 3792 Abstract: We show that non-obtuse trapezoids with identical Neumann spectra are congruent up to rigid motions of the plane. The proof is based on heat trace invariants and some new wave trace invariants associated to certain diffractive billiard trajectories. We use the method of reflections to express the Dirichlet and Neumann wave kernels in terms of the wave kernel of the double polygon. Using Hillairet’s trace formulas for isolated diffractive geodesics and one-parameter families of regular geodesics with geometrically diffractive boundaries for Euclidean surfaces with conical singularities (Hillairet in J Funct Anal 226(1):48–89, 2005), we obtain the new wave trace invariants for trapezoids. To handle the reflected term, we use another result of Hillairet (J Funct Anal 226(1):48–89, 2005), which gives a Fourier integral operator representation for the Keller and Friedlander parametrix (Keller in Proc Symp Appl Math 8:27–52, 1958; Friedlander in Math Proc Camb Philos Soc 90(2):335–341, 1981) of the wave propagator near regular diffractive geodesics. The reason we can only treat the Neumann case is that the wave trace is “more singular” for the Neumann case compared to the Dirichlet case. This is a new observation which is of independent interest. PubDate: 2017-11-08 DOI: 10.1007/s00023-017-0617-7 Issue No:Vol. 18, No. 12 (2017)

Authors:Adam Sawicki; Katarzyna Karnas Pages: 3515 - 3552 Abstract: We consider the problem of deciding if a set of quantum one-qudit gates \(\mathcal {S}=\{g_1,\ldots ,g_n\}\subset G\) is universal, i.e. if \({<}\mathcal {S}{>}\) is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in \(\mathcal {S}\) we assign the orthogonal matrix \(\mathrm {Ad}_g\) that is image of g under the adjoint representation \(\mathrm {Ad}:G\rightarrow SO(\mathfrak {g})\) and \(\mathfrak {g}\) is the Lie algebra of G. The necessary condition for the universality of \(\mathcal {S}\) is that the only matrices that commute with all \(\mathrm {Ad}_{g_i}\) ’s are proportional to the identity. If in addition there is an element in \({<}\mathcal {S}{>}\) whose Hilbert–Schmidt distance from the centre of G belongs to \(]0,\frac{1}{\sqrt{2}}[\) , then \(\mathcal {S}\) is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps and formulate a general classification theorem. PubDate: 2017-11-01 DOI: 10.1007/s00023-017-0604-z Issue No:Vol. 18, No. 11 (2017)

Authors:Sigmund Selberg; Achenef Tesfahun Pages: 3553 - 3564 Abstract: Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Grujić and Kalisch. PubDate: 2017-11-01 DOI: 10.1007/s00023-017-0605-y Issue No:Vol. 18, No. 11 (2017)

Authors:Alan D. Rendall; Juan J. L. Velázquez Pages: 3565 - 3631 Abstract: This paper continues the investigation of the formation of naked singularities in the collapse of collisionless matter initiated in Rendall and Velázquez (Annales Henri Poincaré 12:919–964, 2011). There the existence of certain classes of non-smooth solutions of the Einstein–Vlasov system was proved. Those solutions are self-similar and hence not asymptotically flat. To obtain solutions which are more physically relevant it makes sense to attempt to cut off these solutions in a suitable way so as to make them asymptotically flat. This task, which turns out to be technically challenging, will be carried out in this paper. PubDate: 2017-11-01 DOI: 10.1007/s00023-017-0607-9 Issue No:Vol. 18, No. 11 (2017)

Authors:Michael J. Cole; Juan A. Valiente Kroon Pages: 3651 - 3693 Abstract: We describe the construction of a geometric invariant characterising initial data for the Kerr–Newman spacetime. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr–Newman initial data, and so characterises this type of data. We first illustrate the characterisation of the Kerr–Newman spacetime in terms of Killing spinors. The space-spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar analysis in the vacuum case, we study the properties of solutions to the approximate Killing spinor equation and use them to construct the geometric invariant. PubDate: 2017-11-01 DOI: 10.1007/s00023-017-0606-x Issue No:Vol. 18, No. 11 (2017)

Authors:Wolfgang Wieland Pages: 3695 - 3717 Abstract: In this article, we study the quantum theory of gravitational boundary modes on a null surface. These boundary modes are given by a spinor and a spinor-valued two-form, which enter the gravitational boundary term for self-dual gravity. Using a Fock representation, we quantise the boundary fields and show that the area of a two-dimensional cross section turns into the difference of two number operators. The spectrum is discrete, and it agrees with the one known from loop quantum gravity with the correct dependence on the Barbero–Immirzi parameter. No discrete structures (such as spin network functions, or triangulations of space) are ever required—the entire derivation happens at the level of the continuum theory. In addition, the area spectrum is manifestly Lorentz invariant. PubDate: 2017-11-01 DOI: 10.1007/s00023-017-0598-6 Issue No:Vol. 18, No. 11 (2017)

Authors:Adrian P. C. Lim Pages: 3719 - 3735 Abstract: A hyperlink is a finite set of non-intersecting simple closed curves in \(\mathbb {R} \times \mathbb {R}^3\) . Let S be an orientable surface in \(\mathbb {R}^3\) . The dynamical variables in general relativity are the vierbein e and a \(\mathfrak {su}(2)\times \mathfrak {su}(2)\) -valued connection \(\omega \) . Together with Minkowski metric, e will define a metric g on the manifold. Denote \(A_S(e)\) as the area of S, for a given choice of e. The Einstein–Hilbert action \(S(e,\omega )\) is defined on e and \(\omega \) . We will quantize the area of the surface S by integrating \(A_S(e)\) against a holonomy operator of a hyperlink L, disjoint from S, and the exponential of the Einstein–Hilbert action, over the space of vierbeins e and \(\mathfrak {su}(2)\times \mathfrak {su}(2)\) -valued connections \(\omega \) . Using our earlier work done on Chern–Simons path integrals in \(\mathbb {R}^3\) , we will write this infinite dimensional path integral as the limit of a sequence of Chern–Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between L and S. By assigning an irreducible representation of \(\mathfrak {su}(2)\times \mathfrak {su}(2)\) to each component of L, the area operator gives the total net momentum impact on the surface S. PubDate: 2017-11-01 DOI: 10.1007/s00023-017-0600-3 Issue No:Vol. 18, No. 11 (2017)

Authors:Ko Sanders Pages: 3737 - 3756 Abstract: For a massless free scalar field in a globally hyperbolic space-time we compare the global temperature \(T=\beta ^{-1}\) , defined for the \(\beta \) -KMS states \(\omega ^{(\beta )}\) , with the local temperature \(T_{\omega }(x)\) introduced by Buchholz and Schlemmer. We prove the following claims: (1) whenever \(T_{\omega ^{(\beta )}}(x)\) is defined, it is a continuous, monotonically decreasing function of \(\beta \) at every point x. (2) \(T_{\omega }(x)\) is defined when M is ultra-static with compact Cauchy surface and non-trivial scalar curvature \(R\ge 0\) , \(\omega \) is stationary, and a few other assumptions are satisfied. Our proof of (2) relies on the positive mass theorem. We discuss the necessity of its assumptions, providing counter-examples in an ultra-static space-time with non-compact Cauchy surface and \(R<0\) somewhere. Our results suggest that under suitable circumstances (in particular in the absence of acceleration, rotation and violations of the weak energy condition in the background space-time) both notions of temperature provide qualitatively similar information, and hence the Wick square can be used as a local thermometer. PubDate: 2017-11-01 DOI: 10.1007/s00023-017-0603-0 Issue No:Vol. 18, No. 11 (2017)

Authors:Jochen Zahn Abstract: We discuss a free scalar field subject to generalized Wentzell boundary conditions. On the classical level, we prove well posedness of the Cauchy problem and in particular causality. Upon quantization, we obtain a field that may naturally be restricted to the boundary. We discuss the holographic relation between this boundary field and the bulk field. PubDate: 2017-11-11 DOI: 10.1007/s00023-017-0629-3

Authors:Johannes Bausch; Toby Cubitt; Maris Ozols Abstract: We prove that estimating the ground state energy of a translationally invariant, nearest-neighbour Hamiltonian on a 1D spin chain is \(\textsf {QMA}_{{\textsf {EXP}}}\) -complete, even for systems of low local dimension ( \(\approx 40\) ). This is an improvement over the best previously known result by several orders of magnitude, and it shows that spin-glass-like frustration can occur in translationally invariant quantum systems with a local dimension comparable to the smallest-known non-translationally invariant systems with similar behaviour. While previous constructions of such systems rely on standard models of quantum computation, we construct a new model that is particularly well-suited for encoding quantum computation into the ground state of a translationally invariant system. This allows us to shift the proof burden from optimizing the Hamiltonian encoding a standard computational model, to proving universality of a simple model. Previous techniques for encoding quantum computation into the ground state of a local Hamiltonian allow only a linear sequence of gates, hence only a linear (or nearly linear) path in the graph of all computational states. We extend these techniques by allowing significantly more general paths, including branching and cycles, thus enabling a highly efficient encoding of our computational model. However, this requires more sophisticated techniques for analysing the spectrum of the resulting Hamiltonian. To address this, we introduce a framework of graphs with unitary edge labels. After relating our Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its spectrum by combining matrix analysis and spectral graph theory techniques. PubDate: 2017-10-29 DOI: 10.1007/s00023-017-0609-7

Authors:Misael Avendaño-Camacho; Peter D. Hislop; Carlos Villegas-Blas Abstract: We prove a limiting eigenvalue distribution theorem (LEDT) for suitably scaled eigenvalue clusters around the discrete negative eigenvalues of the hydrogen atom Hamiltonian formed by the perturbation by a weak constant magnetic field. We study the hydrogen atom Zeeman Hamiltonian \({H_V(h,B) = (1/2)( - \imath h {{\nabla }} - {{\mathbf {A}}}(h))^2 - \mathbf{x} ^{-1}}\) , defined on \(L^2 (\mathbb {R}^3)\) , in a constant magnetic field \({\mathbf {B}}(h) = {{\nabla }} \times {{\mathbf {A}}}(h)=(0,0,\epsilon (h)B)\) in the weak field limit \(\epsilon (h) \rightarrow 0\) as \(h\rightarrow {0}\) . We consider the Planck’s parameter h taking values along the sequence \(h=1/(N+1)\) , with \(N=0,1,2,\ldots \) , and \(N\rightarrow \infty \) . We prove a semiclassical \(N \rightarrow \infty \) LEDT of the Szegö-type for the scaled eigenvalue shifts and obtain both (i) an expression involving the regularized classical Kepler orbits with energy \(E=-1/2\) and (ii) a weak limit measure that involves the component \(\ell _3\) of the angular momentum vector in the direction of the magnetic field. This LEDT extends results of Szegö-type for eigenvalue clusters for bounded perturbations of the hydrogen atom to the Zeeman effect. The new aspect of this work is that the perturbation involves the unbounded, first-order, partial differential operator \(w(h, B) = \frac{(\epsilon (h)B)^2}{8} (x_1^2 + x_2^2) - \frac{ \epsilon (h)B}{2} hL_3 ,\) where the operator \(hL_3\) is the third component of the usual angular momentum operator and is the quantization of \(\ell _3\) . The unbounded Zeeman perturbation is controlled using localization properties of both the hydrogen atom coherent states \({\Psi _{{\varvec{\alpha }},N}}\) , and their derivatives \({L_3(h)\Psi _{{\varvec{\alpha }},N}}\) , in the large quantum number regime \(N\rightarrow \infty \) . PubDate: 2017-10-26 DOI: 10.1007/s00023-017-0618-6

Authors:Jinho Baik; Ji Oon Lee Abstract: In Theorem 1.4 (iii) of the original article, we stated that 1 $$\begin{aligned} \sqrt{N} (F_N - F(\beta )) \Rightarrow \mathcal {N}(0, \alpha _2) \end{aligned}$$ in the ferromagnetic regime \(J > 1\) and \(\beta > \frac{1}{2J}\) . The proof was based on Theorem 1.5 (iii), which we proved in the paper, and a known random matrix theory result, given in the second part of (1.19) which reads 2 $$\begin{aligned} N^{1/2} \left( \mu _1 - \left( J + \frac{1}{J}\right) \right) \Rightarrow \mathcal {N}\left( 0, 2\left( 1- \frac{1}{J^2}\right) \right) \end{aligned}$$ for \(J>1\) . PubDate: 2017-10-25 DOI: 10.1007/s00023-017-0613-y

Authors:Davide Ravotti Abstract: Given a compact surface \(\mathcal {M}\) with a smooth area form \(\omega \) , we consider an open and dense subset of the set of smooth closed 1-forms on \(\mathcal {M}\) with isolated zeros which admit at least one saddle loop homologous to zero and we prove that almost every element in the former induces a mixing flow on each minimal component. Moreover, we provide an estimate of the speed of the decay of correlations for smooth functions with compact support on the complement of the set of singularities. This result is achieved by proving a quantitative version for the case of finitely many singularities of a theorem by Ulcigrai (Ergod Theory Dyn Syst 27(3):991–1035, 2007), stating that any suspension flow with one asymmetric logarithmic singularity over almost every interval exchange transformation is mixing. In particular, the quantitative mixing estimate we prove applies to asymmetric logarithmic suspension flows over rotations, which were shown to be mixing by Sinai and Khanin. PubDate: 2017-10-25 DOI: 10.1007/s00023-017-0619-5

Authors:Christoph Richard; Nicolae Strungaru Abstract: We show that the diffraction formula for regular model sets and the Poisson Summation Formula for the underlying lattice can be derived from one another. This is achieved using Fourier analysis of unbounded Radon measures on locally compact abelian groups, as developed by Argabright and de Lamadrid. We also discuss related diffraction results for certain classes of non-regular so-called weak model sets. PubDate: 2017-10-23 DOI: 10.1007/s00023-017-0620-z

Authors:Dejan Gajic Abstract: We consider solutions to the linear wave equation in the interior region of extremal Kerr black holes. We show that axisymmetric solutions can be extended continuously beyond the Cauchy horizon and, moreover, that if we assume suitably fast polynomial decay in time along the event horizon, their local energy is finite. We also extend these results to non-axisymmetric solutions on slowly rotating extremal Kerr–Newman black holes. These results are the analogues of results obtained in Gajic (Commun Math Phys 353(2), 717–770, 2017) for extremal Reissner–Nordström and stand in stark contrast to previously established results for the subextremal case, where the local energy was shown to generically blow up at the Cauchy horizon. PubDate: 2017-10-22 DOI: 10.1007/s00023-017-0614-x

Authors:Andrew Bulawa Abstract: We consider four-dimensional vacuum spacetimes which admit a nonvanishing spacelike Killing field. The quotient with respect to the Killing action is a three-dimensional quotient spacetime (M, g). We establish several results regarding maximal hypersurfaces (spacelike hypersurfaces of zero mean curvature) in such quotient spacetimes. First, we show that a complete noncompact maximal hypersurface must either be a cylinder \(S^1 \times {\mathbb {R}}\) with flat metric or else conformal to the Euclidean plane \({\mathbb {R}}^2\) . Second, we establish positivity of mass for certain maximal hypersurfaces, referring to a analogue of ADM mass adapted for the quotient setting. Finally, while lapse functions corresponding to the maximal hypersurface gauge are necessarily bounded in the four-dimensional asymptotically Euclidean setting, we show that nontrivial quotient spacetimes admit the maximal hypersurface gauge only with unbounded lapse. PubDate: 2017-10-20 DOI: 10.1007/s00023-017-0610-1

Authors:Michele Correggi; Marco Falconi Abstract: We study the quasi-classical limit of a quantum system composed of finitely many nonrelativistic particles coupled to a quantized field in Nelson-type models. We prove that, as the field becomes classical and the corresponding degrees of freedom are traced out, the effective Hamiltonian of the particles converges in resolvent sense to a self-adjoint Schrödinger operator with an additional potential, depending on the state of the field. Moreover, we explicitly derive the expression of such a potential for a large class of field states and show that, for certain special sequences of states, the effective potential is trapping. In addition, we prove convergence of the ground-state energy of the full system to a suitable effective variational problem involving the classical state of the field. PubDate: 2017-10-17 DOI: 10.1007/s00023-017-0612-z

Authors:Giulia Basti; Alessandro Teta Abstract: We consider a three-particle quantum system in dimension three composed of two identical fermions of mass one and a different particle of mass m. The particles interact via two-body short range potentials. We assume that the Hamiltonians of all the two-particle subsystems do not have bound states with negative energy and, moreover, that the Hamiltonians of the two subsystems made of a fermion and the different particle have a zero-energy resonance. Under these conditions and for \(m<m_* = (13.607)^{-1}\) , we give a rigorous proof of the occurrence of the Efimov effect, i.e., the existence of infinitely many negative eigenvalues for the three-particle Hamiltonian H. More precisely, we prove that for \(m>m_*\) the number of negative eigenvalues of H is finite and for \(m<m_*\) the number N(z) of negative eigenvalues of H below \(z<0\) has the asymptotic behavior \(N(z) \sim \mathcal {C}(m) \log z \) for \(z \rightarrow 0^-\) . Moreover, we give an upper and a lower bound for the positive constant \(\mathcal {C}(m)\) . PubDate: 2017-10-14 DOI: 10.1007/s00023-017-0608-8

Authors:Sylvain Lacroix; Benoît Vicedo Abstract: Let \(\mathfrak {g}\) be a semisimple Lie algebra over \(\mathbb {C}\) . Let \(\nu \in \hbox {Aut}\, \mathfrak {g}\) be a diagram automorphism whose order divides \(T \in \mathbb {Z}_{\ge 1}\) . We define cyclotomic \(\mathfrak {g}\) -opers over the Riemann sphere \(\mathbb {P}^1\) as gauge equivalence classes of \(\mathfrak {g}\) -valued connections of a certain form, equivariant under actions of the cyclic group \(\mathbb {Z}/ T\mathbb {Z}\) on \(\mathfrak {g}\) and \(\mathbb {P}^1\) . It reduces to the usual notion of \(\mathfrak {g}\) -opers when \(T = 1\) . We also extend the notion of Miura \(\mathfrak {g}\) -opers to the cyclotomic setting. To any cyclotomic Miura \(\mathfrak {g}\) -oper \(\nabla \) , we associate a corresponding cyclotomic \(\mathfrak {g}\) -oper. Let \(\nabla \) have residue at the origin given by a \(\nu \) -invariant rational dominant coweight \(\check{\lambda }_0\) and be monodromy-free on a cover of \(\mathbb {P}^1\) . We prove that the subset of all cyclotomic Miura \(\mathfrak {g}\) -opers associated with the same cyclotomic \(\mathfrak {g}\) -oper as \(\nabla \) is isomorphic to the \(\vartheta \) -invariant subset of the full flag variety of the adjoint group G of \(\mathfrak {g}\) , where the automorphism \(\vartheta \) depends on \(\nu \) , T and \(\check{\lambda }_0\) . The big cell of the latter is isomorphic to \(N^\vartheta \) , the \(\vartheta \) -invariant subgroup of the unipotent subgroup PubDate: 2017-10-13 DOI: 10.1007/s00023-017-0616-8

Authors:Michele Cirafici Abstract: A large class of \({\mathcal {N}}=2\) quantum field theories admits a BPS quiver description, and the study of their BPS spectra is then reduced to a representation theory problem. In such theories the coupling to a line defect can be modeled by framed quivers. The associated spectral problem characterizes the line defect completely. Framed BPS states can be thought of as BPS particles bound to the defect. We identify the framed BPS degeneracies with certain enumerative invariants associated with the moduli spaces of stable quiver representations. We develop a formalism based on equivariant localization to compute explicitly such BPS invariants, for a particular choice of stability condition. Our framework gives a purely combinatorial solution to this problem. We detail our formalism with several explicit examples. PubDate: 2017-10-13 DOI: 10.1007/s00023-017-0611-0