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:Brian Allen Pages: 1283 - 1306 Abstract: We study the stability of the positive mass theorem (PMT) and the Riemannian Penrose inequality (RPI) in the case where a region of an asymptotically flat manifold \(M^3\) can be foliated by a smooth solution of inverse mean curvature flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically flat manifolds \(U_T^i\subset M_i^3\) , foliated by a smooth solution to IMCF which is uniformly controlled, and if \(\partial U_T^i = \Sigma _0^i \cup \Sigma _T^i\) and \(m_H(\Sigma _T^i) \rightarrow 0\) then \(U_T^i\) converges to a flat annulus with respect to \(L^2\) metric convergence. If instead \(m_H(\Sigma _T^i)-m_H(\Sigma _0^i) \rightarrow 0\) and \(m_H(\Sigma _T^i) \rightarrow m >0\) then we show that \(U_T^i\) converges to a topological annulus portion of the Schwarzschild metric with respect to \(L^2\) metric convergence. PubDate: 2018-04-01 DOI: 10.1007/s00023-017-0641-7 Issue No:Vol. 19, No. 4 (2018)

Authors:Séverin Charbonnier; Bertrand Eynard; François David Abstract: 2D quantum gravity is the idea that a set of discretized surfaces (called map, a graph on a surface), equipped with a graph measure, converges in the large size limit (large number of faces) to a conformal field theory (CFT), and in the simplest case to the simplest CFT known as pure gravity, also known as the gravity dressed (3,2) minimal model. Here, we consider the set of planar Strebel graphs (planar trivalent metric graphs) with fixed perimeter faces, with the measure product of Lebesgue measure of all edge lengths, submitted to the perimeter constraints. We prove that expectation values of a large class of observables indeed converge toward the CFT amplitudes of the (3,2) minimal model. PubDate: 2018-03-24 DOI: 10.1007/s00023-018-0662-x

Authors:Tobias Kuna; Dimitrios Tsagkarogiannis Abstract: We prove absolute convergence of the multi-body correlation functions as a power series in the density uniformly in their arguments. This is done by working in the context of the cluster expansion in the canonical ensemble and by expressing the correlation functions as the derivative of the logarithm of an appropriately extended partition function. In the thermodynamic limit, due to combinatorial cancellations, we show that the coefficients of the above series are expressed by sums over some class of two-connected graphs. Furthermore, we prove the convergence of the density expansion of the “direct correlation function” which is based on a completely different approach and it is valid only for some integral norm. Precisely, this integral norm is suitable to derive the Ornstein–Zernike equation. As a further outcome, we obtain a rigorous quantification of the error in the Percus–Yevick approximation. PubDate: 2018-02-24 DOI: 10.1007/s00023-018-0655-9

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:Dmitry Ostrovsky Abstract: A theory of intermittency differentiation is developed for a general class of 1D infinitely divisible multiplicative chaos measures. The intermittency invariance of the underlying infinitely divisible field is established and utilized to derive a Feynman–Kac equation for the distribution of the total mass of the limit measure by considering a stochastic flow in intermittency. The resulting equation prescribes the rule of intermittency differentiation for a general functional of the total mass and determines the distribution of the total mass and its dependence structure to the first order in intermittency. A class of non-local functionals of the limit measure extending the total mass is introduced and shown to be invariant under intermittency differentiation making the computation of the full high-temperature expansion of the total mass distribution possible in principle. For application, positive integer moments and covariance structure of the total mass are considered in detail. PubDate: 2018-02-19 DOI: 10.1007/s00023-018-0656-8

Authors:Péter Bálint; Péter Nándori; Domokos Szász; Imre Péter Tóth Abstract: We prove exponential correlation decay in dispersing billiard flows on the 2-torus assuming finite horizon and lack of corner points. With applications aimed at describing heat conduction, the highly singular initial measures are concentrated here on 1-dimensional submanifolds (given by standard pairs) and the observables are supposed to satisfy a generalized Hölder continuity property. The result is based on the exponential correlation decay bound of Baladi et al. (Invent Math, 211:39–117, 2018. https://doi.org/10.1007/s00222-017-0745-1) obtained for Hölder continuous observables in these billiards. The model dependence of the bounds is also discussed. PubDate: 2018-02-17 DOI: 10.1007/s00023-018-0648-8

Authors:Laurent Charles; Leonid Polterovich Abstract: We discuss a link between symplectic displacement energy, a fundamental notion of symplectic topology, and the quantum speed limit, a universal constraint on the speed of quantum-mechanical processes. The link is provided by the quantum-classical correspondence formalized within the framework of the Berezin–Toeplitz quantization. PubDate: 2018-02-16 DOI: 10.1007/s00023-018-0649-7

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

Authors:Wei-Xi Li Abstract: In this paper, we consider the Witten Laplacian on 0-forms and give sufficient conditions under which the Witten Laplacian admits a compact resolvent. These conditions are imposed on the potential itself, involving the control of high-order derivatives by lower ones, as well as the control of the positive eigenvalues of the Hessian matrix. This compactness criterion for resolvent is inspired by the one for the Fokker–Planck operator. Our method relies on the nilpotent group techniques developed by Helffer–Nourrigat (Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Birkhäuser Boston Inc., Boston, 1985). PubDate: 2018-02-16 DOI: 10.1007/s00023-018-0659-5

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