COMPLEX KOHN VARIATIONAL PRINCIPLE FOR 2-NUCLEON BOUND-STATE AND SCATTERING WITH THE TENSOR POTENTIAL

Complex Kohn variational principle is applied to the numerical solution of the fully off-shell Lippmann-Schwinger equation for nucleon-nucleon scattering for various partial waves including the coupled S-3(1), D-3(1), channel. Analytic expressions are obtained for all the integrals in the method for a suitable choice of expansion functions. Calculations with the partial waves S-1(0), P-1(1), D-1(2), and S-3(1)-D-3(1) of the Reid soft core potential show that the method converges faster than other solution schemes not only for the phase shift but also for the off-shell t matrix elements. We also show that it is trivial to modify this variational principle in order to make it suitable for bound-state calculation. The bound-state approach is illustrated for the S-3(1)-D-3(1) channel of the Reid soft-core potential for calculating the deuteron binding, wave function, and the D state asymptotic parameters. (c) 1995 Academic Press, Inc.

Renormalization in non-relativistic quantum mechanics

The importance and usefulness of renormalization are emphasized in non-relativistic quantum mechanics. The momentum space treatment of both two-body bound state and scattering problems involving some potentials singular at the origin exhibits ultraviolet divergence. The use of renormalization techniques in these problems leads to finite converged results for both the exact and perturbative solutions. The renormalization procedure is carried out for the quantum two-body problem in different partial waves for a minimal potential possessing only the threshold behaviour and no form factors. The renormalized perturbative and exact solutions for this problem are found to be consistent with each other. The useful role of the renormalization group equations for this problem is also pointed out.

The interaction of D mesons and nucleons at finite density

We present results on the the influence of changes in the masses and sizes of D mesons and nucleons on elastic DN scattering cross sections and phase shifts in a hadronic medium composed of confined quarks in nucleons. We evaluate the changes of the hadronic masses due to changes of the light constituent quarks at finite baryon density using a chiral quark model based on Coulomb gauge QCD. The model contains a confining Coulomb potential and a transverse hyperfine interaction consistent with a finite gluon propagator in the infrared. We present results for the total cross section and the s-wave phase shift at low energies for isospin I=1-for I=0 and other partial waves the results are similar.

Ab-initio determination of x-ray absorption near edge structure (xanes) spectra in an ultrasoft and norm conserving pseudopotentials scheme

X-ray absorption spectroscopy (XAS) is a powerful means of investigation of structural and electronic properties in condensed -matter physics. Analysis of the near edge part of the XAS spectrum, the so – called X-ray Absorption Near Edge Structure (XANES), can typically provide the following information on the photoexcited atom: - Oxidation state and coordination environment. - Speciation of transition metal compounds. - Conduction band DOS projected on the excited atomic species (PDOS). Analysis of XANES spectra is greatly aided by simulations; in the most common scheme the multiple scattering framework is used with the muffin tin approximation for the scattering potential and the spectral simulation is based on a hypothetical, reference structure. This approach has the advantage of requiring relatively little computing power but in many cases the assumed structure is quite different from the actual system measured and the muffin tin approximation is not adequate for low symmetry structures or highly directional bonds. It is therefore very interesting and justified to develop alternative methods. In one approach, the spectral simulation is based on atomic coordinates obtained from a DFT (Density Functional Theory) optimized structure. In another approach, which is the object of this thesis, the XANES spectrum is calculated directly based on an ab – initio DFT calculation of the atomic and electronic structure. This method takes full advantage of the real many-electron final wavefunction that can be computed with DFT algorithms that include a core-hole in the absorbing atom to compute the final cross section. To calculate the many-electron final wavefunction the Projector Augmented Wave method (PAW) is used. In this scheme, the absorption cross section is written in function of several contributions as the many-electrons function of the finale state; it is calculated starting from pseudo-wavefunction and performing a reconstruction of the real-wavefunction by using a transform operator which contains some parameters, called partial waves and projector waves. The aim of my thesis is to apply and test the PAW methodology to the calculation of the XANES cross section. I have focused on iron and silicon structures and on some biological molecules target (myoglobin and cytochrome c). Finally other inorganic and biological systems could be taken into account for future applications of this methodology, which could become an important improvement with respect to the multiscattering approach.

Towards a data-driven analysis of hadronic light-by-light scattering

The hadronic light-by-light contribution to the anomalous magnetic moment of the muon was recently analyzed in the framework of dispersion theory, providing a systematic formalism where all input quantities are expressed in terms of on-shell form factors and scattering amplitudes that are in principle accessible in experiment. We briefly review the main ideas behind this framework and discuss the various experimental ingredients needed for the evaluation of one- and two-pion intermediate states. In particular, we identify processes that in the absence of data for doubly-virtual pion–photon interactions can help constrain parameters in the dispersive reconstruction of the relevant input quantities, the pion transition form factor and the helicity partial waves for γ⁎γ⁎→ππ.

Dispersive approach to hadronic light-by-light scattering

Based on dispersion theory, we present a formalism for a model-independent evaluation of the hadronic light-by-light contribution to the anomalous magnetic moment of the muon. In particular, we comment on the definition of the pion pole in this framework and provide a master formula that relates the effect from ππ intermediate states to the partial waves for the process γ * γ * → ππ. All contributions are expressed in terms of on-shell form factors and scattering amplitudes, and as such amenable to an experimental determination.

Roy-Steiner equations for piN scattering

In this talk, we present a coupled system of integral equations for the πN → πN (s-channel) and ππ → N̅N (t-channel) lowest partial waves, derived from Roy–Steiner equations for pion–nucleon scattering. After giving a brief overview of this system of equations, we present the solution of the t-channel sub-problem by means of Muskhelishvili–Omnès techniques, and solve the s-channel sub-problem after finding a set of phase shifts and subthreshold parameters which satisfy the Roy–Steiner equations.

A search for an exotic meson in the gamma + P decaying to delta++ + pi- + eta reaction

A Partial Waves Analysis (PWA) of γp → Δ ++X → pπ+ π - (η) data taken with the CLAS detector at Jefferson Lab is presented in this work. This reaction is of interest because the Δ++ restricts the isospin of the possible X states, leaving the PWA with a smaller combination of partial waves, making it ideal to look for exotic mesons. It was proposed by Isgur and Paton that photoproduction is a plausible source for the Jpc=1–+ state through flux tube excitation. The π1(1400) is such a state that has been produced with the use of hadron production but it has yet to be seen in photoproduction. A mass independent amplitude analysis of this channel was performed, followed by a mass dependent fit to extract the resonance parameters. The procedure used an event-based maximum likelihood method to maintain all correlations in the kinematics. The intensity and phase motion is mapped out for the contributing signals without requiring assumptions about the underlying processes. The strength of the PWA is in the analysis of the phase motion, which for resonance behavior is well defined. In the data presented, the ηπ– invariant mass spectrum shows contributions from the a0(980) and a2(1320) partial waves. No π1 was observed under a clear a2 signal after the angular distributions of the decay products were analyzed using an amplitude analysis. In addition, this dissertation discusses trends in the data, along with the implemented techniques.

Plane waves and spherical means applied to partial differential equations.

Bibliography: p. 165-170.

Models of collective cell motion for cell populations with different aspect ratio : diffusion, proliferation and travelling waves

Continuum, partial differential equation models are often used to describe the collective motion of cell populations, with various types of motility represented by the choice of diffusion coefficient, and cell proliferation captured by the source terms. Previously, the choice of diffusion coefficient has been largely arbitrary, with the decision to choose a particular linear or nonlinear form generally based on calibration arguments rather than making any physical connection with the underlying individual-level properties of the cell motility mechanism. In this work we provide a new link between individual-level models, which account for important cell properties such as varying cell shape and volume exclusion, and population-level partial differential equation models. We work in an exclusion process framework, considering aligned, elongated cells that may occupy more than one lattice site, in order to represent populations of agents with different sizes. Three different idealizations of the individual-level mechanism are proposed, and these are connected to three different partial differential equations, each with a different diffusion coefficient; one linear, one nonlinear and degenerate and one nonlinear and nondegenerate. We test the ability of these three models to predict the population level response of a cell spreading problem for both proliferative and nonproliferative cases. We also explore the potential of our models to predict long time travelling wave invasion rates and extend our results to two dimensional spreading and invasion. Our results show that each model can accurately predict density data for nonproliferative systems, but that only one does so for proliferative systems. Hence great care must be taken to predict density data for with varying cell shape.

Travelling waves for a velocity–jump model of cell migration and proliferation

Cell invasion, characterised by moving fronts of cells, is an essential aspect of development, repair and disease. Typically, mathematical models of cell invasion are based on the Fisher–Kolmogorov equation. These traditional parabolic models can not be used to represent experimental measurements of individual cell velocities within the invading population since they imply that information propagates with infinite speed. To overcome this limitation we study combined cell motility and proliferation based on a velocity–jump process where information propagates with finite speed. The model treats the total population of cells as two interacting subpopulations: a subpopulation of left–moving cells, $L(x,t)$, and a subpopulation of right–moving cells, $R(x,t)$. This leads to a system of hyperbolic partial differential equations that includes a turning rate, $\Lambda \ge 0$, describing the rate at which individuals in the population change direction of movement. We present exact travelling wave solutions of the system of partial differential equations for the special case where $\Lambda = 0$ and in the limit that $\Lambda \to \infty$. For intermediate turning rates, $0 < \Lambda < \infty$, we analyse the travelling waves using the phase plane and we demonstrate a transition from smooth monotone travelling waves to smooth nonmonotone travelling waves as $\Lambda$ decreases through a critical value $\Lambda_{crit}$. We conclude by providing a qualitative comparison between the travelling wave solutions of our model and experimental observations of cell invasion. This comparison indicates that the small $\Lambda$ limit produces results that are consistent with experimental observations.

Finite-amplitude love-waves on an isotropic layered half-space

A pair of semi-linear hyperbolic partial differential equations governing the slow variations in amplitude and phase of a quasi-monochromatic finite-amplitude Love-wave on an isotropic layered half-space is derived using the method of multiple-scales. The analysis of the exact solution of these equations for a signalling problem reveals that the amplitude of the wave remains constant along its characteristic and that the phase of the wave increases linearly behind the wave-front.

Nonlinear mode coupling of surface acoustic waves on an isotropic solid

The nonlinear mode coupling between two co-directional quasi-harmonic Rayleigh surface waves on an isotropic solid is analysed using the method of multiple scales. This procedure yields a system of six semi-linear hyperbolic partial differential equations with the same principal part governing the slow variations in the (complex) amplitudes of the two fundamental, the two second harmonic and the two combination frequency waves at the second stage of the perturbation expansion. A numerical solution of these equations for excitation by monochromatic signals at two arbitrary frequencies, indicates that there is a continuous transfer of energy back and forth among the fundamental, second harmonic and combination frequency waves due to mode coupling. The mode coupling tends to be more pronounced as the frequencies of the interacting waves approach each other.

Quasi-simple wave solutions for acoustic gravity waves

The special class of quasi-simple wave solutions is studied for the system of partial differential equations governing inviscid acoustic gravity waves. It is shown that these traveling wave solutions do not admit shocks. Periodic solutions are found to exist when there is no propagation in the vertical direction. The solutions for some particular cases are depicted graphically. Physics of Fluids is copyrighted by The American Institute of Physics.

Trapped waves in the neighborhood of a sonic-type singularity

A model equation is derived to study trapped nonlinear waves with a turning effect, occurring in disturbances induced on a two-dimensional steady flow. Only unimodal disturbances under the short wave assumption are considered, when the wave front of the induced disturbance is plane. In the neighbourhood of certain special points of sonic-type singularity, the disturbances are governed by a single first-order partial differential equation in two independent variables. The equation depends on the steady flow through three parameters, which are determined by the variations of velocity and depth, for example (in the case of long surface water waves), along and perpendicular to the wave front. These parameters help us to examine various relative effects. The presence of shocks in a continuously accelerating or decelerating flow has been studied in detail.