Authors:Yves Colin de Verdière; Françoise Truc Pages: 1419 - 1438 Abstract: In this paper, we try to put the results of Smilansky et al. on “Topological resonances” on a mathematical basis. A key role in the asymptotic of resonances near the real axis for Quantum Graphs is played by the set of metrics for which there exist compactly supported eigenfunctions. We give several estimates on the dimension of this semi-algebraic set, in particular in terms of the girth of the graph. The case of trees is also discussed. PubDate: 2018-05-01 DOI: 10.1007/s00023-018-0672-8 Issue No:Vol. 19, No. 5 (2018)

Authors:Jean Dolbeault; Maria J. Esteban; Ari Laptev; Michael Loss Pages: 1439 - 1463 Abstract: We prove magnetic interpolation inequalities and Keller–Lieb–Thirring estimates for the principal eigenvalue of magnetic Schrödinger operators. We establish explicit upper and lower bounds for the best constants and show by numerical methods that our theoretical estimates are accurate. PubDate: 2018-05-01 DOI: 10.1007/s00023-018-0663-9 Issue No:Vol. 19, No. 5 (2018)

Authors:Loïc Le Treust; Thomas Ourmières-Bonafos Pages: 1465 - 1487 Abstract: This paper deals with the study of the two-dimensional Dirac operator with infinite mass boundary conditions in sectors. We investigate the question of self-adjointness depending on the aperture of the sector: when the sector is convex it is self-adjoint on a usual Sobolev space, whereas when the sector is non-convex it has a family of self-adjoint extensions parametrized by a complex number of the unit circle. As a by-product of the analysis, we are able to give self-adjointness results on polygonal domains. We also discuss the question of distinguished self-adjoint extensions and study basic spectral properties of the Dirac operator with a mass term in the sector. PubDate: 2018-05-01 DOI: 10.1007/s00023-018-0661-y Issue No:Vol. 19, No. 5 (2018)

Authors:Nikolaos Roidos Pages: 1489 - 1505 Abstract: Let X be a two-dimensional smooth manifold with boundary \(S^{1}\) and \(Y=[1,\infty )\times S^{1}\) . We consider a family of complete surfaces arising by endowing \(X\cup _{S^{1}}Y\) with a parameter-dependent Riemannian metric, such that the restriction of the metric to Y converges to the hyperbolic metric as a limit with respect to the parameter. We describe the associated spectral and scattering theory of the Laplacian for such a surface. We further show that on Y the zero \(S^{1}\) -Fourier coefficient of the generalized eigenfunction of this Laplacian, as a family with respect to the parameter, approximates in a certain sense, for large values of the spectral parameter, the zero \(S^{1}\) -Fourier coefficient of the generalized eigenfunction of the Laplacian for the case of a surface with hyperbolic cusp. PubDate: 2018-05-01 DOI: 10.1007/s00023-018-0669-3 Issue No:Vol. 19, No. 5 (2018)

Authors:Andreas Deuchert; Alissa Geisinger; Christian Hainzl; Michael Loss Pages: 1507 - 1527 Abstract: We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. In the case of vanishing angular momentum, our results carry over to the three-dimensional case. PubDate: 2018-05-01 DOI: 10.1007/s00023-018-0665-7 Issue No:Vol. 19, No. 5 (2018)

Authors:Bas Janssens Pages: 1587 - 1610 Abstract: Generalised spin structures describe spinor fields that are coupled to both general relativity and gauge theory. We classify those generalised spin structures for which the corresponding fields admit an infinitesimal action of the space–time diffeomorphism group. This can be seen as a refinement of the classification of generalised spin structures by Avis and Isham (Commun Math Phys 72:103–118, 1980). PubDate: 2018-05-01 DOI: 10.1007/s00023-018-0667-5 Issue No:Vol. 19, No. 5 (2018)

Authors:Matthias Täufer; Martin Tautenhahn Pages: 1151 - 1165 Abstract: We consider non-ergodic magnetic random Schrödinger operators with a bounded magnetic vector potential. We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of arguments from Klein (Commun Math Phys 323(3):1229–1246, 2013), combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from Borisov et al. (J Math Phys, arXiv:1512.06347 [math.AP]). This generalizes Klein’s result to operators with a bounded magnetic vector potential. Moreover, we study the dependence of the Wegner-constant on the disorder parameter. In particular, we show that above the model-dependent threshold \(E_0(\infty ) \in (0, \infty ]\) , it is impossible that the Wegner-constant tends to zero if the disorder increases. This result is new even for the standard (ergodic) Anderson Hamiltonian without magnetic field. PubDate: 2018-04-01 DOI: 10.1007/s00023-017-0640-8 Issue No:Vol. 19, No. 4 (2018)

Authors:Niels Benedikter; Jérémy Sok; Jan Philip Solovej Pages: 1167 - 1214 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: 2018-04-01 DOI: 10.1007/s00023-018-0644-z Issue No:Vol. 19, No. 4 (2018)

Authors:Ho Lee; Ernesto Nungesser Abstract: In this paper we consider the Einstein–Vlasov system with Bianchi VII \(_0\) symmetry. Under the assumption of small data we show that self-similarity breaking occurs for reflection symmetric solutions. This generalizes the previous work concerning the non-tilted fluid case (Wainwright et al. in Class Quantum Gravity 16:2577–2598, 1999) to the Vlasov case, and we obtain detailed information about the late-time behaviour of metric and matter terms. PubDate: 2018-04-24 DOI: 10.1007/s00023-018-0678-2

Authors:Werner Kirsch; Georgi Raikov Abstract: We consider Schrödinger operators with a random potential which is the square of an alloy-type potential. We investigate their integrated density of states and prove Lifshits tails. Our interest in this type of models is triggered by an investigation of randomly twisted waveguides. PubDate: 2018-04-23 DOI: 10.1007/s00023-018-0680-8

Authors:Eric P. Hanson; Alain Joye; Yan Pautrat; Renaud Raquépas Abstract: We analyse Landauer’s principle for repeated interaction systems consisting of a reference quantum system \(\mathcal {S}\) in contact with an environment \(\mathcal {E}\) which is a chain of independent quantum probes. The system \(\mathcal {S}\) interacts with each probe sequentially, for a given duration, and Landauer’s principle relates the energy variation of \(\mathcal {E}\) and the decrease of entropy of \(\mathcal {S}\) by the entropy production of the dynamical process. We consider refinements of the Landauer bound at the level of the full statistics (FS) associated with a two-time measurement protocol of, essentially, the energy of \(\mathcal {E}\) . The emphasis is put on the adiabatic regime where the environment, consisting of \(T \gg 1\) probes, displays variations of order \(T^{-1}\) between the successive probes, and the measurements take place initially and after T interactions. We prove a large deviation principle and a central limit theorem as \(T \rightarrow \infty \) for the classical random variable describing the entropy production of the process, with respect to the FS measure. In a special case, related to a detailed balance condition, we obtain an explicit limiting distribution of this random variable without rescaling. At the technical level, we obtain a non-unitary adiabatic theorem generalizing that of Hanson et al. (Commun Math Phys 349(1):285–327, 2017) and analyse the spectrum of complex deformations of families of irreducible completely positive trace-preserving maps. PubDate: 2018-04-21 DOI: 10.1007/s00023-018-0679-1

Authors:Alexander C. R. Belton; Michał Gnacik; J. Martin Lindsay Abstract: We give a simple and direct treatment of the strong convergence of quantum random walks to quantum stochastic operator cocycles, via the semigroup decomposition of such cocycles. Our approach also delivers convergence of the pointwise product of quantum random walks to the quantum stochastic Trotter product of the respective limit cocycles, thereby revealing the algebraic structure of the limiting procedure. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described. PubDate: 2018-04-21 DOI: 10.1007/s00023-018-0676-4

Authors:Jean-Philippe Miqueu Abstract: This paper is devoted to the spectral analysis of the magnetic Laplacian with semiclassical parameter \(h>0\) , defined on a bounded and regular domain \(\Omega \) of \(\mathbb {R}^2\) with Neumann magnetic boundary condition, in the case when the magnetic field vanishes along a smooth curve intersecting \(\partial \Omega \) . We investigate the behavior of the eigenvalues and the associated eigenfunctions when the semiclassical parameter h tends to 0. We provide a one term asymptotic of the first eigenvalue as well as a full asymptotic expansion of the bottom of the spectrum as \(h\rightarrow 0\) . PubDate: 2018-04-19 DOI: 10.1007/s00023-018-0681-7

Authors:Alan Carey; Fritz Gesztesy; Jens Kaad; Galina Levitina; Roger Nichols; Denis Potapov; Fedor Sukochev Abstract: We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators \(H_0 = \alpha \cdot (-i \nabla )\) for all space dimensions \(n \in {{\mathbb {N}}}\) , \(n \geqslant 2\) . This is a new result for all dimensions other than three, in particular, it applies to the two-dimensional case which is known to be of some relevance in applications to graphene. We also prove an essential self-adjointness result for first-order matrix-valued differential operators with Lipschitz coefficients. PubDate: 2018-04-17 DOI: 10.1007/s00023-018-0675-5

Authors:Volker Betz; Steffen Dereich; Peter Mörters Abstract: We consider effective models of condensation where the condensation occurs as time t goes to infinity. We provide natural conditions under which the buildup of the condensate occurs on a spatial scale of 1 / t and has the universal form of a Gamma density. The exponential parameter of this density is determined only by the equation and the total mass of the condensate, while the power law parameter may in addition depend on the decay properties of the initial condition near the condensation point. We apply our results to some examples, including simple models of Bose–Einstein condensation. PubDate: 2018-04-17 DOI: 10.1007/s00023-018-0673-7

Authors:Gaku Hoshino Abstract: We study the Cauchy problem for a quadratic system of nonlinear Schrödinger equations in \(L^2\) -setting with the space dimension \(n=1,2\) or 3. Recently, the author showed that the local solution for the system of nonlinear Schrödinger equations has space-time analytic smoothing effect for data with exponentially weighted \(L^2\) -norm. Also as is well known, the quadratic nonlinear Schrödinger equations have global solutions in \(L^2\) -subcritical setting. Our main purpose of this study is to show real analyticity in both space and time variables of the unique global solution with data which has large exponentially weighted \(L^2\) -norm. PubDate: 2018-04-16 DOI: 10.1007/s00023-018-0677-3

Authors:P. J. Forrester; J. R. Ipsen; Dang-Zheng Liu Abstract: We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib–Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix transformation which maps the space of polynomial ensembles onto itself. This matrix transformation is closely related to the so-called hyperbolic Harish-Chandra–Itzykson–Zuber integral. PubDate: 2018-02-24 DOI: 10.1007/s00023-018-0654-x

Authors:Jan Felipe van Diejen; Erdal Emsiz; Ignacio Nahuel Zurrián Abstract: We employ a discrete integral-reflection representation of the double affine Hecke algebra of type \(C^\vee C\) at the critical level \(\text {q}=1\) , to endow the open finite q-boson system with integrable boundary interactions at the lattice ends. It is shown that the Bethe Ansatz entails a complete basis of eigenfunctions for the commuting quantum integrals in terms of Macdonald’s three-parameter hyperoctahedral Hall–Littlewood polynomials. PubDate: 2018-02-23 DOI: 10.1007/s00023-018-0658-6

Authors:L. Castellani; R. Catenacci; P. A. Grassi Abstract: We reformulate super-quantum mechanics in the context of integral forms. This framework allows to interpolate between different actions for the same theory, connected by different choices of picture changing operators (PCO). In this way we retrieve component and superspace actions and prove their equivalence. The PCO are closed integral forms and can be interpreted as super-Poincaré duals of bosonic submanifolds embedded into a supermanifold. We use them to construct Lagrangians that are top integral forms, and therefore can be integrated on the whole supermanifold. The \(D=1, N=1\) and the \(D=1, N=2\) cases are studied, in a flat and in a curved supermanifold. In this formalism, we also consider coupling with gauge fields, Hilbert space of quantum states, and observables. PubDate: 2018-02-23 DOI: 10.1007/s00023-018-0653-y

Authors:András Vasy; Michał Wrochna Abstract: We consider the wave equation on asymptotically Minkowski spacetimes and the Klein–Gordon equation on even asymptotically de Sitter spaces. In both cases, we show that the extreme difference of propagators (i.e., retarded propagator minus advanced, or Feynman minus anti-Feynman), defined as Fredholm inverses, induces a symplectic form on the space of solutions with wave front set confined to the radial sets. Furthermore, we construct isomorphisms between the solution spaces and symplectic spaces of asymptotic data. As an application of this result, we obtain distinguished Hadamard two-point functions from asymptotic data. Ultimately, we prove that non-interacting Quantum Field Theory on asymptotically de Sitter spacetimes extends across the future and past conformal boundary, i.e., to a region represented by two even asymptotically hyperbolic spaces. Specifically, we show this to be true both at the level of symplectic spaces of solutions and at the level of Hadamard two-point functions. PubDate: 2018-02-16 DOI: 10.1007/s00023-018-0650-1