Frank Tabakin

QUASILINEARIZATION APPROACH TO NONLINEAR PROBLEMS IN PHYSICS WITH APPLICATION TO NONLINEAR ODES

Abstract The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of the proof to partial differential equations is straight forward. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to different nonlinear ordinary differential equations in physics, such as the Blasius, Duffing, Lane–Emden and Thomas–Fermi equations. The first few quasilinear iterations already provide extremely accurate and numerically stable answers.

Improved theoretical pion--nucleus optical potentials

Abstract An approach which makes the first order pion-nucleus optical potential theoretically sound is presented. This study should permit higher order improvements to the potential to be more meaningful and the nuclear structure information extracted from pi-nucleus scattering to be more reliable. Based on multiple scattering theory, three optical potentials are constructed and studied in momentum space. These models are the popular Kisslinger potential, the local “Laplacian” potential, and an “improved off-shell potential;” the latter one is derived from absorptive separable pion-nucleon potentials which exactly reproduce on-shell πN scattering. By working in momentum space and explicitly including πN resonances and off-shell effects in the definition of the optical potential, the approach described here is capable of handling any number of pi-nucleon partial waves, is applicable over a very wide energy region, is based on a physical model for off-shell behavior, and is extended easily to include higher order effects. The optical potentials are inserted into two different relativistic wave equations to determine the total cross section and elastic differential cross section for pi-nucleus scattering. It is found that the various models for off-shell πN scattering determine significantly different πC 12 scattering, with the improved off-shell model preferred on theoretical grounds. Also discussed is the importance of properly transforming πN scattering to the pi-nucleus c.m. system, the origin of the shift in the peak position of the π − C total cross section, and the reason for the increased diffractive nature of the differential cross section at 180 MeV.

Completeness rules for spin observables in pseudoscalar meson photoproduction

The number and type of measurements needed to ascertain the amplitudes for pseudoscalar meson photoproduction are analyzed in this paper. It is found that eight carefully selected measurements can determine the four transversity amplitudes without discrete ambiguities. That number of measurements is one less than previously believed. We approach this problem in two distinct ways: (1) solving for the amplitude magnitudes and phases directly, and (2) using a bilinear helicity product formulation to map an algebra of measurements over to the well-known algebra of the 4{times}4 gamma matrices. It is shown that the latter method leads to an alternate proof that eight carefully chosen experiments suffice for determining the transversity amplitudes completely. In addition, Fierz transformations of the gamma matrices are used to develop useful linear and nonlinear relationships between the spin observables. These relationships not only help in finding complete sets of experiments, but also yield important constraints between the 16 observables for this reaction. {copyright} {ital 1997} {ital The American Physical Society}

Nuclear saturation and the smoothness of nucleon-nucleon potentials

Abstract The two-nucleon and nuclear matter problems are solved by matrix inversion in momentum space. Direct matrix inversion of the Lippman-Schwinger and Brueckner equations is shown to be useful for general nuclear potentials including ones that are local, nonlocal, weak, strong, central, or noncentral. This flexibility is employed to study the relationship between nuclear saturation and the smoothness of the two-nucleon interaction. Five potentials are considered that give approximately equivalent phase shifts but differ in their smoothness. Two examples of smooth potentials with very weak nonlocal tensor terms are given. The potentials are classified according to their smoothness by calculating the wave function defect and the wound integral for each case. The binding energy of nuclear matter is calculated for each potential using the effective mass and angle-averaged Pauli operator approximations. A self-consistent hole spectrum and a free particle spectrum are used. A systematic dependence of saturation on the smoothness of the two-nucleon interaction is found. Only strong potentials with strong tensor terms yield correct saturation, whereas very smooth potentials produce overbinding and large equilibrium densities.

Charged pion photoproduction from light nuclei

Abstract The photoproduction of 0–150-MeV charged pions from light nuclei is studied from a distorted wave impulse approximation (DWIA) approach. The final nuclear states are restricted to a finite set of isospin analogs of excited states of the target nucleus. The final state interactions of the pion with the residual nucleus are incorporated via optical potentials. The elementary photoproduction operator used is that of Blomqvist and Laget which is derived in a general reference frame. To gain insight into the predictive power of this DWIA approach, total and differential cross sections for π ± production from 6 Li, 7 Li, 10 B, 12 C, and 14 N are calculated and compared with available data. It is found that, with a few exceptions, reasonable agreement is obtained between theory and experiment as long as the nuclear wave functions are constrained to fit other electromagnetic and weak processes and the optical potentials are constrained to fit pion-nucleus elastic scattering data. We conclude that, at this stage, using the Blomqvist-Laget operator in a DWIA calculation adequately describes the dynamics of charged pion photoproduction from complex nuclei. We illustrate how this reaction can be used to obtain information on the short range nature of the pion wave function and on nuclear wave functions. Shortcomings of and improvements on this calculation are also suggested.

Dynamical coupled-channels approach to hadronic and electromagnetic kaon-hyperon production on the proton

A dynamical coupled-channels formalism for processes

Mixing of the f(0) and a(0) scalar mesons in threshold photoproduction

\ensuremath{\pi}N\ensuremath{\rightarrow}\mathit{KY}

Momentum-space study of pion-nucleus inelastic scattering☆

and

QDENSITY—A Mathematica quantum computer simulation☆

\ensuremath{\gamma}N\ensuremath{\rightarrow}\mathit{KY}

Pseudoscalar meson photoproduction: From known to undiscovered resonances

is presented that provides a comprehensive investigation of recent data on the