Persistence of Anderson localization in Schr\"odinger operators with decaying random potentials

We show persistence of both Anderson and dynamical localization in Schr\"odinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schr\"odinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than $|x|^{-2}$ at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as $|x|^{-\alpha}$ at infinity, we determine the number of bound states below a given energy $E<0$, asymptotically as $\alpha\downarrow 0$. To show that bound states located at the bottom of the spectrum are related to the phenomenon of Anderson localization in the corresponding ergodic model, we prove: (a) these states are exponentially localized with a localization length that is uniform in the decay exponent $\alpha$; (b)~ dynamical localization holds uniformly in $\alpha$

Effect of housing enrichment and type of flooring on the performance and behaviour of female rabbits

[EN] This study investigated the effect of housing enrichments (scratching card, gnawing material and a platform), of a change in height and in the type of flooring on the live weight, reproductive performance and behaviour of female rabbits, as well as on the feed intake and spatial distribution of females and their kits. A total of 40 multiparous female rabbits were monitored in three consecutive reproductive cycles (48-d intervals). Four days before parturition in each reproductive cycle, the females were randomly assigned to one of the five types of housing: Control (CNT: 102×47×30 cm, L×W× H); Scratching card (SCT: containing a scratching card); Gnawing materials (GNW: CNT dimensions plus a compressed lucerne hay block and a wooden stick); Platform (PLT: 102×47×60 cm, including a platform with a plastic floor) and Combination (CBN: PLT dimensions with the scratching card, the gnawing materials and a platform). Data were only recorded during the first and third reproductive cycles. The living conditions did not significantly alter the females’ live weight (4889 g at housing; 4890 g at mid-lactation; 4867 g at weaning), reproductive performance (9.0 kits born alive), survival of the kits (90%), nor the feed intake of females and their litters (542 g/day). Providing animals with a gnawing block stimulated Gnawing behaviour (median frequency per group: CTL=0.00, SCT=0.00, GNW=4.69, PLT=0.00, and CBN=2.34; PRearing up behaviour (median frequencies per group: CTL=0.00, SCT=0.00, GNW=0.00, PLT=2.08, and CBN=3.12; P=0.06), and when a platform was present, the rabbits used it (mean values per group: CTL=0.00, SCT=0.00, GNW=0.00, PLT=1.79, and CBN=4.91; P=0.003). Regarding the type of floor, females appeared to prefer the plastic mesh flooring (31.2%) to the wire mesh flooring (18.8%). To sum up, providing female rabbits with simple enrichments appears to stimulate specific behaviours like Gnawing and Rearing up and may contribute to their wellbeing.Huang, Y.; Bréda, J.; Savietto, D.; Debrusse, A.; Bonnemère, J.; Gidenne, T.; Combes, S.... (2021). Effect of housing enrichment and type of flooring on the performance and behaviour of female rabbits. World Rabbit Science. 29(4):275-285. https://doi.org/10.4995/wrs.2021.15848OJS275285294Altmann J. 1974. Observational study of behavior: sampling methods. Behaviour, 49: 227-266. https://doi.org/10.1163/156853974X00534Animal Welfare Committee. 2009. Five Freedoms. Available at https://webarchive.nationalarchives.gov.uk/20110909181150/http://www.fawc.org.uk/freedoms.htm. Accessed February 2021.ANSES. 2018. ANSES proposes a definition of animal welfare and sets the foundation for its research and expert appraisal work. 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Qingdao, China. 1: 663-666.Gerencsér Z., Farkas T.P., Dal Bosco A., Filiou E., Matics Z., Odermatt M., Paci G., Szendrő Z. 2016. The usage of multilevel platforms in growing rabbits housed in large pens as affected by platfor mmaterial (wire-mesh vs plastic-mesh). In Proc. 11th World Rabbit Congress, 15-18 June, 2016. Qingdao, China. 1: 671-674.Hawkins P., Hubrecht R., Buckwell A., Cubitt S., Howard B., Jackson A., Poirier, G. M. 2008. Refining rabbit care. A resource for those working with rabbits in research. RSPCA, West Sussex and UFAW, Hertfordshire.Huang Y., Breda J., Savietto D., Debrusse A., Combes S., Fortun-Lamothe L. 2021. Part-time grouping of rabbit does in enriched housing: effects on performances, injury occurrence and enrichment use. Animal. in press.Jordan D., Gorjanc G., Kermauner A., Štuhec I. 2011. The behaviour of individually housed growing rabbits and the influence of gnawing sticks as environmental enrichment on daily rhythm of behavioural patterns duration. Acta Agriculturae Slovenica, 98: 51-61.Jordan D., Gorjanc G., Štuhec I. 2008. Wooden sticks as environmental enrichment: effect on fattening and carcass traits of individually housed growing rabbits. World Rabbit Sci., 16: 237-243. https://doi.org/10.4995/wrs.2008.619Lang C., Hoy S. 2011. Investigations on the use of an elevated platform in group cages by growing rabbits. World Rabbit Sci., 19: 95-101. https://doi.org/10.4995/wrs.2011.800Leach M.C., Allweiler S., Richardson C., Roughan J.V., Narbe R., Flecknell P.A. 2009. Behavioural effects of ovariohysterectomy and oral administration of meloxicam in laboratory housed rabbits. Res. Vet. Sci., 87: 336-347. https://doi.org/10.1016/j.rvsc.2009.02.001Luzi F., Ferrante V., Heinzl E. Verga M. 2003. Effect of environmental enrichment on productive performance and welfare aspects in fattening rabbits. Ital. J. Anim. Sci., 2: 438-440.Matics Z., Farkas T. P., Dal Bosco A., Szendrő Z., Filiou E., Nagy I., Odermatt M., Paci G., Gerencsér Z. 2018. Comparison of pens without and with multilevel platforms for growing rabbits. Ital. J. Anim. Sci., 17: 469-476. https://doi.org/10.1080/1828051X.2017.1363640Mikó A., Matics Z., Gerencsér Z., Odermatt M., Radnai I., Nagy I., Szendrő K. Szendrő, Z. 2014. Performance and welfare of rabbit does in various caging systems. Animal, 8: 1146-1152. https://doi.org/10.1017/S1751731114001244Mirabito L., Buthon L., Cialdi G., Galliot P., Souchet C. 1999. Effet du logement des lapines en cages réhaussées avec plate-forme: Premiers resultats. In Proc.: 8èmes Journées de la Recherche Cunicole. 9-10 June, 1999. Paris, France. 1: 67-70.Mirabito L., Galliot P., Souchet C. 2000. Effect of different ways of cage enrichment on the productive traits and mortality of fattening rabbits. In Proc.: 7th World Rabbit Congress 4-7 July, 2000. Valencia, Spain. 1: 4-7.Monclús R., Rödel H. G. 2008. 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Asymptotic behaviour of the spectrum of a waveguide with distant perturbations

We consider the waveguide modelled by a $n$-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators ''localized'' in a certain sense, and the distance between their ''supports'' tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such system. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some particular examples of the distant perturbations. The examples are the potential, second order differential operator, magnetic Schroedinger operator, curved and deformed waveguide, delta interaction, and integral operator

Generalized eigenvalue-counting estimates for the Anderson model

We generalize Minami's estimate for the Anderson model and its extensions to $n$ eigenvalues, allowing for $n$ arbitrary intervals and arbitrary single-site probability measures with no atoms. As an application, we derive new results about the multiplicity of eigenvalues and Mott's formula for the ac-conductivity when the single site probability distribution is H\"older continuous.Comment: Minor revisio

Widths of the Hall Conductance Plateaus

We study the charge transport of the noninteracting electron gas in a two-dimensional quantum Hall system with Anderson-type impurities at zero temperature. We prove that there exist localized states of the bulk order in the disordered-broadened Landau bands whose energies are smaller than a certain value determined by the strength of the uniform magnetic field. We also prove that, when the Fermi level lies in the localization regime, the Hall conductance is quantized to the desired integer and shows the plateau of the bulk order for varying the filling factor of the electrons rather than the Fermi level.Comment: 94 pages, v2: a revision of Sec. 5; v3: an error in Sec. 7 is corrected, major revisions of Sec. 7 and Appendix E, Sec. 7 is enlarged to Secs. 7-12, minor corrections; v4: major revisions, accepted for publication in Journal of Statistical Physics; v5: minor corrections, accepted versio

Eta-Helium Quasi-Bound States

The cross section and tensor analysing power t_20 of the d\vec{d}->eta 4He reaction have been measured at six c.m. momenta, 10 < p(eta) < 90 MeV/c. The threshold value of t_20 is consistent with 1/\sqrt{2}, which follows from parity conservation and Bose symmetry. The much slower momentum variation observed for the reaction amplitude, as compared to that for the analogous pd->eta 3He case, suggests strongly the existence of a quasi-bound state in the eta-4He system and optical model fits indicate that this probably also the case for eta-3He.Comment: LaTeX, uses elsart.sty, 10 pages, 3 Postscript figures, Submitted to Physics Letters

Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method

A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated by the sign-indefinite single-site potential, which is however sign-definite at the boundary of its support. For this class of Anderson operators we establish a finite-volume criterion which implies that above mentioned the fractional moment decay property holds. This constructive criterion is satisfied at typical perturbative regimes, e. g. at spectral boundaries which satisfy 'Lifshitz tail estimates' on the density of states and for sufficiently strong disorder. We also show how the fractional moment method facilitates the proof of exponential (spectral) localization for such random potentials.Comment: 29 pages, 1 figure, to appear in AH

On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum

We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E_n~n^\alpha, with 0<\alpha<1. In particular, the gaps between successive eigenvalues decay as n^{\alpha-1}. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate |V(t)_{m,n}|0, p>=1 and \gamma=(1-\alpha)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and \epsilon is small enough. More precisely, for any initial condition \Psi\in Dom(H^{1/2}), the diffusion of energy is bounded from above as _\Psi(t)=O(t^\sigma) where \sigma=\alpha/(2\ceil{p-1}\gamma-1/2). As an application we consider the Hamiltonian H(t)=|p|^\alpha+\epsilon*v(\theta,t) on L^2(S^1,d\theta) which was discussed earlier in the literature by Howland

Reconciling MOND and dark matter?

Observations of galaxies suggest a one-to-one analytic relation between the inferred gravity of dark matter at any radius and the enclosed baryonic mass, a relation summarized by Milgrom's law of modified Newtonian dynamics (MOND). However, present-day covariant versions of MOND usually require some additional fields contributing to the geometry, as well as an additional hot dark matter component to explain cluster dynamics and cosmology. Here, we envisage a slightly more mundane explanation, suggesting that dark matter does exist but is the source of MOND-like phenomenology in galaxies. We assume a canonical action for dark matter, but also add an interaction term between baryonic matter, gravity, and dark matter, such that standard matter effectively obeys the MOND field equation in galaxies. We show that even the simplest realization of the framework leads to a model which reproduces some phenomenological predictions of cold dark matter (CDM) and MOND at those scales where these are most successful. We also devise a more general form of the interaction term, introducing the medium density as a new order parameter. This allows for new physical effects which should be amenable to observational tests in the near future. Hence, this very general framework, which can be furthermore related to a generalized scalar-tensor theory, opens the way to a possible unification of the successes of CDM and MOND at different scales.Comment: 9 page

Adiabatic non-equilibrium steady states in the partition free approach

Consider a small sample coupled to a finite number of leads, and assume that the total (continuous) system is at thermal equilibrium in the remote past. We construct a non-equilibrium steady state (NESS) by adiabatically turning on an electrical bias between the leads. The main mathematical challenge is to show that certain adiabatic wave operators exist, and to identify their strong limit when the adiabatic parameter tends to zero. Our NESS is different from, though closely related with the NESS provided by the Jak{\v s}i{\'c}-Pillet-Ruelle approach. Thus we partly settle a question asked by Caroli {\it et al} in 1971 regarding the (non)equivalence between the partitioned and partition-free approaches