Eigenvalue bounds for a class of singular potentials in N dimensions

The eigenvalue bounds obtained earlier [J. Phys. A: Math. Gen. 31 (1998) 963] for smooth transformations of the form V(x) = g(x^2) + f(1/x^2) are extended to N-dimensions. In particular a simple formula is derived which bounds the eigenvalues for the spiked harmonic oscillator potential V(x) = x^2 + lambda/x^alpha, alpha > 0, lambda > 0, and is valid for all discrete eigenvalues, arbitrary angular momentum ell, and spatial dimension N.Comment: 10 pages (plain tex with 2 ps figures). J.Phys.A:Math.Gen.(In Press

Spectral characteristics for a spherically confined -1/r + br^2 potential

We consider the analytical properties of the eigenspectrum generated by a class of central potentials given by V(r) = -a/r + br^2, b>0. In particular, scaling, monotonicity, and energy bounds are discussed. The potential $V(r)$ is considered both in all space, and under the condition of spherical confinement inside an impenetrable spherical boundary of radius R. With the aid of the asymptotic iteration method, several exact analytic results are obtained which exhibit the parametric dependence of energy on a, b, and R, under certain constraints. More general spectral characteristics are identified by use of a combination of analytical properties and accurate numerical calculations of the energies, obtained by both the generalized pseudo-spectral method, and the asymptotic iteration method. The experimental significance of the results for both the free and confined potential V(r) cases are discussed.Comment: 16 pages, 4 figure

Taming the Yukawa potential singularity: improved evaluation of bound states and resonance energies

Using the tools of the J-matrix method, we absorb the 1/r singularity of the Yukawa potential in the reference Hamiltonian, which is handled analytically. The remaining part, which is bound and regular everywhere, is treated by an efficient numerical scheme in a suitable basis using Gauss quadrature approximation. Analysis of resonance energies and bound states spectrum is performed using the complex scaling method, where we show their trajectories in the complex energy plane and demonstrate the remarkable fact that bound states cross over into resonance states by varying the potential parameters.Comment: 8 pages, 2 tables, 1 figure. 2 mpg videos and 1 pdf table file are available upon request from the corresponding Autho

Comparative power demand of mechanical and aeration imposed shear in an immersed membrane bioreactor

The power demanded for the application of mechanically-imposed shear on an immersed flat sheet (iFS) membrane bioreactor (MBR) has been compared to that of conventional membrane air scouring. Literature correlations based on the Ostwald model were used to define the rheological characteristics of an MBR sludge. The correlation of specific power demand (, in Watts per m2 membrane area) with shear rate γ in s-1 was developed from first principles through a consideration of the force balance on the system in the case of mechanically-imposed shear. The corresponding aeration imposed shear correlation was interpreted from literature information. The analysis revealed the energy required to impose a shear mechanically through oscillation (or reciprocation) of the membrane to be between 20 and 70% less than that demanded for providing the same shear by conventional aeration of the immersed membrane. The energy saving increases with decreasing shear in accordance with a power demand ratio (aeration:mechanical) of 1400γ-1.4 for a specific sludge rheology. Whilst the absolute value is dependent on the sludge rheology, the aeration:mechanical power demand ratio is determined by the difference in the two exponents in the respective correlations between and γ. Consequently, aeration-imparted shear becomes energetically favoured beyond some threshold shear rate value (∼180 s-1, based on the boundary conditions applied in the current study). The outcomes qualitatively corroborate findings from the limited practical measurement of energy demand in MBRs fitted with reciprocating immersed membranes

Energies and wave functions for a soft-core Coulomb potential

For the family of model soft Coulomb potentials represented by V(r) = -\frac{Z}{(r^q+\beta^q)^{\frac{1}{q}}}, with the parameters Z>0, \beta>0, q \ge 1, it is shown analytically that the potentials and eigenvalues, E_{\nu\ell}, are monotonic in each parameter. The potential envelope method is applied to obtain approximate analytic estimates in terms of the known exact spectra for pure power potentials. For the case q =1, the Asymptotic Iteration Method is used to find exact analytic results for the eigenvalues E_{\nu\ell} and corresponding wave functions, expressed in terms of Z and \beta. A proof is presented establishing the general concavity of the scaled electron density near the nucleus resulting from the truncated potentials for all q. Based on an analysis of extensive numerical calculations, it is conjectured that the crossing between the pair of states [(\nu,\ell),(\nu',\ell')], is given by the condition \nu'\geq (\nu+1) and \ell' \geq (\ell+3). The significance of these results for the interaction of an intense laser field with an atom is pointed out. Differences in the observed level-crossing effects between the soft potentials and the hydrogen atom confined inside an impenetrable sphere are discussed.Comment: 13 pages, 5 figures, title change, minor revision

Clogging vs. fouling in immersed membrane bioreactors

Whilst the fouling of MBR membrane surfaces has been very extensively explored by the academic community, there is an increasingly widespread recognition by practitioners of the issue of clogging of membrane channels with sludge solids, sometimes termed “sludging”. The study undertaken has quantified this phenomenon using a bespoke test cell allowing a flat sheet membrane channel to be viewed directly during operation and the accumulated solids determined by digital image processing. Sludging behaviour has then been correlated both with the sludge properties, from sludge samples taken from both an industrial and municipal MBR, and the permeability decline rate data. The work has revealed the expected trends in fouling propensity, as quantified by the exponent n of the Δp/Δt = m.exp(nJ) correlation from classical flux-step tests. With zero membrane aeration the industrial samples exhibited sludging, the filling of the complete thickness of the membrane channel with sludge solids, whereas for municipal sludge the solids formed a cake layer which did not fill the channel. In the absence of sludging the permeability decline followed the expected pattern of increasing at the elevated soluble COD and capillary suction time values of the industrial sludge, compared with municipal sludge at the same solids concentration range (8–12 g.L−1). However, there was no evident correlation between fouling (permeability decline without sludging) and sludging: incipient sludging did not appear to influence permeability, though can be assumed to negatively impact on long-term operation, or relate to the sCOD concentration. Sludging instead appeared to depend on the sludge physical properties, and primarily the viscosity: sludge samples at high viscosities were found to exhibit a different air-scour pattern to that at normal MLSS concentrations. Outcomes suggest that sludging is caused by rheological conditions promoting bubble coalescence and bubble stream constriction, reducing the exposure of the membrane surface to scouring air

Memory of the Unjamming Transition during Cyclic Tiltings of a Granular Pile

Discrete numerical simulations are performed to study the evolution of the micro-structure and the response of a granular packing during successive loading-unloading cycles, consisting of quasi-static rotations in the gravity field between opposite inclination angles. We show that internal variables, e.g., stress and fabric of the pile, exhibit hysteresis during these cycles due to the exploration of different metastable configurations. Interestingly, the hysteretic behaviour of the pile strongly depends on the maximal inclination of the cycles, giving evidence of the irreversible modifications of the pile state occurring close to the unjamming transition. More specifically, we show that for cycles with maximal inclination larger than the repose angle, the weak contact network carries the memory of the unjamming transition. These results demonstrate the relevance of a two-phases description -strong and weak contact networks- for a granular system, as soon as it has approached the unjamming transition.Comment: 13 pages, 15 figures, soumis \`{a} Phys. Rev.

Theory and computation of directional nematic phase ordering

A computational study of morphological instabilities of a two-dimensional nematic front under directional growth was performed using a Landau-de Gennes type quadrupolar tensor order parameter model for the first-order isotropic/nematic transition of 5CB (pentyl-cyanobiphenyl). A previously derived energy balance, taking anisotropy into account, was utilized to account for latent heat and an imposed morphological gradient in the time-dependent model. Simulations were performed using an initially homeotropic isotropic/nematic interface. Thermal instabilities in both the linear and non-linear regimes were observed and compared to past experimental and theoretical observations. A sharp-interface model for the study of linear morphological instabilities, taking into account additional complexity resulting from liquid crystalline order, was derived. Results from the sharp-interface model were compared to those from full two-dimensional simulation identifying the specific limitations of simplified sharp-interface models for this liquid crystal system. In the nonlinear regime, secondary instabilities were observed to result in the formation of defects, interfacial heterogeneities, and bulk texture dynamics.Comment: first revisio

Coherent states associated to the wavefunctions and the spectrum of the isotonic oscillator

Classes of coherent states are presented by replacing the labeling parameter $z$ of Klauder-Perelomov type coherent states by confluent hypergeometric functions with specific parameters. Temporally stable coherent states for the isotonic oscillator Hamiltonian are presented and these states are identified as a particular case of the so-called Mittag-Leffler coherent states.Comment: 12 page