Authors:C. Cedzich; T. Geib; F. A. Grünbaum; C. Stahl; L. Velázquez; A. H. Werner; R. F. Werner Pages: 325 - 383 Abstract: We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition. No translation invariance is assumed. The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm-continuous path along which all the mentioned properties remain valid. Of the three indices, two are related to the asymptotic behavior far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. The results apply to the Hamiltonian case as well. In this case, all compact perturbations can be contracted, so the third index is not defined. Our classification extends the one known in the translation- invariant case, where the asymptotic right and left indices add up to zero, and the third one vanishes, leaving effectively only one independent index. When two translation-invariant bulks with distinct indices are joined, the left and right asymptotic indices of the joined walk are thereby fixed, and there must be eigenvalues at 1 or \(-\,1\) (bulk-boundary correspondence). Their location is governed by the third index. We also discuss how the theory applies to finite lattices, with suitable homogeneity assumptions. PubDate: 2018-02-01 DOI: 10.1007/s00023-017-0630-x Issue No:Vol. 19, No. 2 (2018)

Authors:Horia D. Cornean; Valeriu Moldoveanu; Claude-Alain Pillet Pages: 411 - 442 Abstract: The main goal of this paper is to put on solid mathematical grounds the so-called non-equilibrium Green’s function transport formalism for open systems. In particular, we derive the Jauho–Meir–Wingreen formula for the time-dependent current through an interacting sample coupled to non-interacting leads. Our proof is non-perturbative and uses neither complex-time Keldysh contours nor Langreth rules of ‘analytic continuation.’ We also discuss other technical identities (Langreth, Keldysh) involving various many-body Green’s functions. Finally, we study the Dyson equation for the advanced/retarded interacting Green’s function and we rigorously construct its (irreducible) self-energy, using the theory of Volterra operators. PubDate: 2018-02-01 DOI: 10.1007/s00023-017-0638-2 Issue No:Vol. 19, No. 2 (2018)

Authors:Aldo Procacci; Benedetto Scoppola; Elisabetta Scoppola Pages: 443 - 462 Abstract: We present a simple one-dimensional Ising-type spin system on which we define a completely asymmetric Markovian single spin-flip dynamics. We study the system at a very low, yet nonzero, temperature, and we show that for free boundary conditions the Gibbs measure is stationary for such dynamics, while introducing in a single site a \(+\) condition the stationary measure changes drastically, with macroscopical effects. We achieve this result defining an absolutely convergent series expansion of the stationary measure around the zero temperature system. Interesting combinatorial identities are involved in the proofs. PubDate: 2018-02-01 DOI: 10.1007/s00023-017-0627-5 Issue No:Vol. 19, No. 2 (2018)

Authors:Sylvain Lacroix; Benoît Vicedo Pages: 71 - 139 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 ... PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0616-8 Issue No:Vol. 19, No. 1 (2018)

Authors:Jochen Zahn Pages: 163 - 187 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: 2018-01-01 DOI: 10.1007/s00023-017-0629-3 Issue No:Vol. 19, No. 1 (2018)

Authors:Michele Correggi; Marco Falconi Pages: 189 - 235 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: 2018-01-01 DOI: 10.1007/s00023-017-0612-z Issue No:Vol. 19, No. 1 (2018)

Authors:Rui Han; Chris A. Marx Pages: 249 - 265 Abstract: We quantify the coupling asymptotics for the Lyapunov exponent of a one-frequency quasi-periodic Schrödinger operator with analytic potential sampling function. The result refines the well-known lower bound of the Lyapunov exponent by Sorets and Spencer. PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0626-6 Issue No:Vol. 19, No. 1 (2018)

Authors:Jan P. Boroński; Jiří Kupka; Piotr Oprocha Pages: 267 - 281 Abstract: Motivated by a recent result of Ciesielski and Jasiński we study periodic point free Cantor systems that are conjugate to systems with vanishing derivative everywhere, and more generally locally radially shrinking maps. Our study uncovers a whole spectrum of dynamical behaviors attainable for such systems, providing new counterexamples to the Conjecture of Edrei from 1952, first disproved by Williams in 1954. PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0623-9 Issue No:Vol. 19, No. 1 (2018)

Authors:Gianfausto Dell’Antonio; Alessandro Michelangeli; Raffaele Scandone; Kenji Yajima Pages: 283 - 322 Abstract: We prove that, for arbitrary centres and strengths, the wave operators for three-dimensional Schrödinger operators with multi-centre local point interactions are bounded in \(L^p({\mathbb {R}}^3)\) for \(1<p<3\) and unbounded otherwise. PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0628-4 Issue No:Vol. 19, No. 1 (2018)

Authors:Jeremiah Birrell; Jan Wehr Abstract: We study the dynamics of an inertial particle coupled to forcing, dissipation, and noise in the small mass limit. We derive an expression for the limiting (homogenized) joint distribution of the position and (scaled) velocity degrees of freedom. In particular, weak convergence of the joint distributions is established, along with a bound on the convergence rate for a wide class of expected values. PubDate: 2018-02-06 DOI: 10.1007/s00023-018-0646-x

Authors:Niels Benedikter; Jérémy Sok; Jan Philip Solovej Abstract: The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics, Bose–Einstein condensation and quantum chemistry. We reformulate the Dirac–Frenkel approximation principle in terms of reduced density matrices and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov–de Gennes and Hartree–Fock–Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov–de Gennes equations in energy space and discuss conserved quantities. PubDate: 2018-01-24 DOI: 10.1007/s00023-018-0644-z

Authors:Christian D. Jäkel; Jens Mund Abstract: We establish the Haag–Kastler axioms for a class of interacting quantum field theories on the two-dimensional de Sitter space, which satisfy finite speed of light. The \({\mathscr {P}} (\varphi )_2\) model constructed in [3], describing massive scalar bosons with polynomial interactions, provides an example. PubDate: 2018-01-20 DOI: 10.1007/s00023-018-0647-9

Authors:Felix Finster; Alexander Strohmaier Abstract: In Section 5.1 in [1] it is incorrectly claimed that condition (A) is equivalent to the vanishing of the operator B in the expansion. PubDate: 2017-12-28 DOI: 10.1007/s00023-017-0632-8

Authors:Wojciech Dybalski; Alessandro Pizzo Abstract: Let \(H_{P,\sigma }\) be the single-electron fiber Hamiltonians of the massless Nelson model at total momentum P and infrared cut-off \(\sigma >0\) . We establish detailed regularity properties of the corresponding n-particle ground state wave functions \(f^n_{P,\sigma }\) as functions of P and \(\sigma \) . In particular, we show that $$\begin{aligned} \ \ \partial _{P^j}f^{n}_{P,\sigma }(k_1,\ldots , k_n) , \ \ \partial _{P^j} \partial _{P^{j'}} f^{n}_{P,\sigma }(k_1,\ldots , k_n) \!\le \! \frac{1}{\sqrt{n!}} \frac{(c\lambda _0)^n}{\sigma ^{\delta _{\lambda _0}}} \prod _{i=1}^n\frac{ \chi _{[\sigma ,\kappa )}(k_i)}{ k_i ^{3/2}}, \end{aligned}$$ where c is a numerical constant, \(\lambda _0\mapsto \delta _{\lambda _0}\) is a positive function of the maximal admissible coupling constant which satisfies \(\lim _{\lambda _0\rightarrow 0}\delta _{\lambda _0}=0\) and \(\chi _{[\sigma ,\kappa )}\) is the (approximate) characteristic function of the energy region between the infrared cut-off \(\sigma \) and the ultraviolet cut-off \(\kappa \) . While the analysis of the first derivative is relatively straightforward, the second derivative requires a new strategy. By solving a non-commutative recurrence relation, we derive a novel formula for \(f^n_{P,\sigma }\) with improved infrared properties. In this representation \(\partial _{P^{j'}}\partial _{P^{j}}f^n_{P,\sigma }\) is amenable to sharp estimates obtained by iterative analytic perturbation theory in part II of this series of papers. The bounds stated above are instrumental for scattering theory of two electrons in the Nelson model, as explained in part I of this series. PubDate: 2017-12-28 DOI: 10.1007/s00023-017-0642-6

Authors:Xinliang An; Xuefeng Zhang Abstract: The vacuum Einstein equations in \(5+1\) dimensions are shown to admit solutions describing naked singularity formation in gravitational collapse from nonsingular asymptotically locally flat initial data that contain no trapped surface. We present a class of specific examples with topology \(\mathbb {R}^{3+1} \times S^2\) . Thanks to the Kaluza–Klein dimensional reduction, these examples are constructed by lifting continuously self-similar solutions of the 4-dimensional Einstein-scalar field system with a negative exponential potential. The latter solutions are obtained by solving a 3-dimensional autonomous system of first-order ordinary differential equations with a combined analytic and numerical approach. Their existence provides a new test-bed for weak cosmic censorship in higher-dimensional gravity. In addition, we point out that a similar attempt of lifting Christodoulou’s naked singularity solutions of massless scalar fields fails to capture formation of naked singularities in \(4+1\) dimensions, due to a diverging Kretschmann scalar in the initial data. PubDate: 2017-12-20 DOI: 10.1007/s00023-017-0631-9

Authors:Gerhard Bräunlich; David Hasler; Markus Lange Abstract: We consider expansions of eigenvalues and eigenvectors of models of quantum field theory. For a class of models known as generalized spin–boson model, we prove the existence of asymptotic expansions of the ground state and the ground state energy to arbitrary order. We need a mild but very natural infrared assumption, which is weaker than the assumption usually needed for other methods such as operator theoretic renormalization to be applicable. The result complements previously shown analyticity properties. PubDate: 2017-12-19 DOI: 10.1007/s00023-017-0625-7

Authors:Peter Hintz Abstract: We show that a stationary solution of the Einstein–Maxwell equations which is close to a non-degenerate Reissner–Nordström–de Sitter solution is in fact equal to a slowly rotating Kerr–Newman–de Sitter solution. The proof uses the nonlinear stability of the Kerr–Newman–de Sitter family of black holes with small angular momenta, recently established by the author, together with an extension argument for Killing vector fields. Our black hole uniqueness result only requires the solution to have high but finite regularity; in particular, we do not make any analyticity assumptions. PubDate: 2017-12-15 DOI: 10.1007/s00023-017-0633-7

Authors:Matthias Christandl; M. Burak Şahinoğlu; Michael Walter Abstract: We prove that the asymptotic behavior of the recoupling coefficients of the symmetric group \(S_k\) is characterized by a quantum marginal problem: they decay polynomially in k if there exists a quantum state of three particles with given eigenvalues for their reduced density operators and exponentially otherwise. As an application, we deduce solely from symmetry considerations of the coefficients the strong subadditivity property of the von Neumann entropy, first proved by Lieb and Ruskai (J Math Phys 14:1938–1941, 1973). Our work may be seen as a non-commutative generalization of the representation-theoretic aspect of the recently found connection between the quantum marginal problem and the Kronecker coefficient of the symmetric group, which has applications in quantum information theory and algebraic complexity theory. This connection is known to generalize the correspondence between Weyl’s problem on the addition of Hermitian matrices and the Littlewood–Richardson coefficients of SU(d). In this sense, our work may also be regarded as a generalization of Wigner’s famous observation of the semiclassical behavior of the recoupling coefficients (here also known as 6j or Racah coefficients), which decay polynomially whenever a tetrahedron with given edge lengths exists. More precisely, we show that our main theorem contains a characterization of the possible eigenvalues of partial sums of Hermitian matrices thus presenting a representation-theoretic characterization of a generalization of Weyl’s problem. The appropriate geometric objects to SU(d) recoupling coefficients are thus tuples of Hermitian matrices and to \(S_k\) recoupling coefficients they are three-particle quantum states. PubDate: 2017-12-15 DOI: 10.1007/s00023-017-0639-1

Authors:David Klein; Jake Reschke Abstract: Robertson–Walker spacetimes within a large class are geometrically extended to larger cosmologies that include spacetime points with zero and negative cosmological times. In the extended cosmologies, the big bang is lightlike, and though singular, it inherits some geometric structure from the original spacetime. Spacelike geodesics are continuous across the cosmological time zero submanifold which is parameterized by the radius of Fermi space slices, i.e., by the proper distances along spacelike geodesics from a comoving observer to the big bang. The continuous extension of the metric, and the continuously differentiable extension of the leading Fermi metric coefficient \(g_{\tau \tau }\) of the observer, restrict the geometry of spacetime points with pre-big bang cosmological time coordinates. In our extensions the big bang is two dimensional in a certain sense, consistent with some findings in quantum gravity. PubDate: 2017-12-14 DOI: 10.1007/s00023-017-0634-6

Authors:Raphaël Belliard; Bertrand Eynard; Olivier Marchal Abstract: To any flat section equation of the form \(\nabla _0\Psi =\Phi \Psi \) in a principal bundle over a Riemann surface ( \(\nabla _0\) is a reference connection), we associate an infinite sequence of “correlators”, symmetric n-differentials on \(\Sigma \) that we denote \(\{W-n\}_{n \in \mathcal {N}}\) . The goal of this article is to prove that these correlators are solutions to “loop equations,” the same ones satisfied by correlation functions in random matrix models, or equivalently Ward identities of Virasoro or \({\mathcal {W}}\) -symmetric CFT. PubDate: 2017-12-02 DOI: 10.1007/s00023-017-0622-x