ALBERT

Description:    We prove optimal high-frequency resolvent estimates for self-adjoint operators of the form G =( i Ñ + b ( x )) 2 + V ( x ) on L 2 ( R n ), n ³ 3 , where the magnetic potential b ( x ) and the electric potential V ( x ) are long-range and large. As an application, we prove dispersive estimates for the wave group e it Ö G in the case n  = 3 for potentials b ( x ), V ( x ) =  O (| x | −2- δ ) for | x | 〉〉 1 , where δ  〉 0. Content Type Journal Article Pages 1-23 DOI 10.1007/s00023-012-0178-8 Authors Fernando Cardoso, Departamento de Matemática, Universidade Federal de Pernambuco, CEP 50540-740 Recife, PE, Brazil Claudio Cuevas, Departamento de Matemática, Universidade Federal de Pernambuco, CEP 50540-740 Recife, PE, Brazil Georgi Vodev, Département de Mathématiques, Université de Nantes, UMR 6629 du CNRS, 2, rue de la Houssinière, BP 92208, 44332 Nantes Cedex 03, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We prove a new lower bound on the indirect Coulomb energy in two-dimensional quantum mechanics in terms of the single particle density of the system. The new universal lower bound is an alternative to the Lieb–Solovej–Yngvason bound with a smaller constant, C = (4/3) 3/2 Ö   5 p - 1   » 5.90 〈 C LSY = 192 Ö   2 p   » 481.27 , which also involves an additive gradient energy term of the single particle density. Content Type Journal Article Pages 1-12 DOI 10.1007/s00023-012-0176-x Authors Rafael D. Benguria, Departmento de Física, P. Universidad Católica de Chile, Santiago, Chile Pablo Gallegos, Departmento de Física, P. Universidad Católica de Chile, Santiago, Chile Matěj Tušek, Departmento de Física, P. Universidad Católica de Chile, Santiago, Chile Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider a two-dimensional magnetic Schrödinger operator on a square lattice with a spatially stationary random magnetic field. We prove Anderson localization near the spectral edges. We use a new approach to establish a Wegner estimate that does not rely on the monotonicity of the energy on the random parameters. Content Type Journal Article Pages 1-13 DOI 10.1007/s00023-012-0177-9 Authors László Erdős, Institute of Mathematics, University of Munich, Theresienstr. 39, 80333 Munich, Germany David Hasler, Institute of Mathematics, University of Munich, Theresienstr. 39, 80333 Munich, Germany Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider translation-invariant quantum systems in thermodynamic limit. We argue that their energy-momentum spectra should have shapes consistent with effective models involving quasiparticles. Our main example is second quantized homogeneous interacting Fermi gas in a large cubic box with periodic boundary conditions, at zero temperature. We expect that its energy-momentum spectrum has a positive energy gap and a positive critical velocity. Content Type Journal Article Pages 1-36 DOI 10.1007/s00023-012-0185-9 Authors Jan Dereziński, Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoża 74, 00-682 Warsaw, Poland Krzysztof A. Meissner, Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Hoża 69, 00-681 Warsaw, Poland Marcin Napiórkowski, Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoża 74, 00-682 Warsaw, Poland Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    The thermodynamic limit of the internal energy and the entropy of the system of quantum interacting particles in random medium is shown to exist under the crucial requirements of stability and temperedness of interactions. The energy turns out to be proportional to the number of particles and/or volume of the system in the thermodynamic limit. The obtained results require very general assumptions on the random one-particle model. The methods are mainly based on subadditive type of inequalities. Content Type Journal Article Pages 1-32 DOI 10.1007/s00023-012-0186-8 Authors Nikolaj A. Veniaminov, Laboratoire Analyse Géométrie et Applications, Université Paris 13 Nord, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider the Schrödinger operator on \mathbb R 2 with a locally square-integrable periodic potential V and give an upper bound for the Bethe–Sommerfeld threshold (the minimal energy above which no spectral gaps occur) with respect to the square-integrable norm of V on a fundamental domain, provided that V is small. As an application, we prove the spectrum of the two-dimensional Schrödinger operator with the Poisson type random potential almost surely equals the positive real axis or the whole real axis, according as the negative part of the single-site potential equals zero or not. The latter result completes the missing part of the result by Ando et al. (Ann Henri Poincaré 7:145–160, 2006 ). Content Type Journal Article Pages 1-26 DOI 10.1007/s00023-012-0180-1 Authors Masahiro Kaminaga, Department of Electrical Engineering and Information Technology, Tohoku Gakuin University, Tagajo, 985-8537 Japan Takuya Mine, Graduate School of Science and Technology, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, 606-8585 Japan Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider the physical model of a classical mechanical system (called “small system”) undergoing repeated interactions with a chain of identical small pieces (called “environment”). This physical setup constitutes an advantageous way of implementing dissipation for classical systems; it is at the same time Hamiltonian and Markovian. This kind of model has already been studied in the context of quantum mechanical systems, where it was shown to give rise to quantum Langevin equations in the limit of continuous time interactions (Attal and Pautrat in Ann Henri Poincaré 7:59–104, 2006 ), but it has never been considered for classical mechanical systems yet. The aim of this article is to compute the continuous limit of repeated interactions for classical systems and to prove that they give rise to particular stochastic differential equations (SDEs) in the limit. In particular, we recover the usual Langevin equations associated with the action of heat baths. In order to obtain these results, we consider the discrete-time dynamical system induced by Hamilton’s equations and the repeated interactions. We embed it into a continuous-time dynamical system and compute the limit when the time step goes to 0. This way, we obtain a discrete-time approximation of SDE, considered as a deterministic dynamical system on the Wiener space, which is not exactly of the usual Euler scheme type. We prove the L p and almost sure convergence of this scheme. We end up with applications to concrete physical examples such as a charged particle in a uniform electric field or a harmonic interaction. We obtain the usual Langevin equation for the action of a heat bath when considering a damped harmonic oscillator as the small system. Content Type Journal Article Pages 1-42 DOI 10.1007/s00023-012-0179-7 Authors Julien Deschamps, C.N.R.S., Institut Camille Jordan, Université de Lyon, Université de Lyon 1, 21, av Claude Bernard, 69622 Villeurbanne Cedex, France Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    For stationary, barotropic fluids in Newtonian gravity we give simple criteria on the equation of state and the “law of motion” which guarantee finite or infinite extent of the fluid region (providing a priori estimates for the corresponding stationary Newton–Euler system). Under more restrictive conditions, we can also exclude the presence of “hollow” configurations. Our main result, which does not assume axial symmetry, uses the virial theorem as the key ingredient and generalises a known result in the static case. In the axially symmetric case stronger results are obtained and examples are discussed. Content Type Journal Article Pages 1-19 DOI 10.1007/s00023-012-0181-0 Authors Patryk Mach, M. Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland Walter Simon, M. Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We consider the spectral theory and inverse scattering problem for discrete Schrödinger operators on the hexagonal lattice. We give a procedure for reconstructing finitely supported potentials from the scattering matrices for all energies. The same procedure is applicable for the inverse scattering problem on the triangle lattice. Content Type Journal Article Pages 1-37 DOI 10.1007/s00023-012-0183-y Authors Kazunori Ando, Graduate School of Pure and Applied Science, University of Tsukuba, Tennoudai 1-1-1, Tsukuba, Ibaraki, 305-8571 Japan Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637

Description:    We prove persistence of absolutely continuous spectrum for the Anderson model on a general class of tree-like graphs. Content Type Journal Article Pages 1-23 DOI 10.1007/s00023-011-0139-7 Authors Florina Halasan, Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637