Manuel Rotenberg

Application of sturmian functions to the Schroedinger three-body problem: Elastic e+-H scattering☆

Abstract The properties of a set of eigenfunctions closely related to Schroedinger wave functions are exploited to deal with certain three-body problems. These functions, termed Sturmian functions, have a distinct advantage over Schroedinger functions when used as an expansion basis since they form a complete set without a continuum, regardless of the potential existing between the particles. This enables one to examine, for example, the mechanism of polarization of the deuteron in elastic N-D scattering. Further use of the Sturmian functions can be made in describing elastic scattering of electrons and positrons from hydrogen. A detailed calculation of elastic scattering of positrons from atomic hydrogen is carried out. It is shown that Sturmian functions converge rapidly when they are used as an expansion basis, and provide phase shifts which are slightly greater than rigorous lower bounds provided by a variational treatment.

Theory and Application of Sturmian Functions

Publisher Summary This chapter focuses on scattering problems that have made significant use of Sturmians. The use of Sturmians to include effects of the continuum has the appeal of a systematic development, but there are other ways to include the continuum. One advantage of pseudo-states over Sturmians is that they are applicable to inelastic scattering problems, but by using linear combinations of Sturmians the same end can be achieved. The subsequent interpretation of the phase shifts in terms of elastic scattering cross sections remains unaltered from the usual hydrogenic treatment. The appeal of the Sturmians, however, lies in their ability to describe continuum admixtures to the closed channels, thus their usefulness can be retained if they are used only for this purpose. Elastic scattering from a two-particle system in its ground state occurs between the lowest two of these points; between the second and third points an inelastic channel is open, another inelastic channel opens between the third and fourth points.

Transport theory for growing cell populations

The partial differential equation that describes the growth of cell populations whose maturation rate is random is developed. The equation resembles that used in classical transport theory but mitotic boundary conditions and the restriction of the maturation rate to non-negative values brings out new features and new problems. This is a generalization of a previously published formulation in which cells could make transitions at random between only two maturation velocities: a characteristic velocity and zero. Growth rates, cycle time distributions and pulsed labeled mitotic curves are calculated for a simple choice of parameters. A numerical algorithm that is suited to the solution of the transport equation is given.

Theory of population transport

A transport equation for population systems is developed. The equation is for a distribution function in which position, age and time are independent variables. It includes as special cases the classical equations of Malthus, Verhulst and Volterra. It is shown under what conditions diffusion-like behavior of migration result, and how mass-migration terms come about. For simplicity, the discussion is confined to a single species; the generalization to interacting species is mentioned briefly.

Equilibrium and stability in populations whose interactions are age-specific

The continuous equations for a closed set of interacting populations are examined with regard to equilibrium and stability. The novel feature that is introduced is the age-specificity of the interactions between populations as well as the age-specificity of the effects of limited resources. First, single self-limiting populations are studied. Next, a set of interacting populations is examined for equilibrium points. Finally, the formal theory of stability in an age-specific description is detailed. Numerous simple examples are given.

Theory of distributed quiescent state in the cell cycle.

A generalization of the familiar two-compartment or G0 model of the cell cycle is described. Instead of reserving the quiescent state strictly to newly born cells, it is distributed throughout the cell cycle. A cell may cease its proliferative activities anywhere in the cycle with a probability depending on its maturity. The probability of returning to cycle is also a function of maturity. Analytical expressions for cycle time distributions, growth rates, wave frequency and relative damping rates are derived for certain cases. A stable, diffusion-free numerical algorithm is used to work out some examples.

Effect of certain stochastic parameters on extinction and harvested populations

The logistic equation was originally intended to describe only the growth of a population as a function of time. However, when it is used to trace the time development of a population in an environment with a fluctuating carrying capacity, the equation must describe both its growth and diminution. It is suggested that it may not be realistic to use the same rate constant for both an increasing and decreasing population since reproduction rates and death rates are independent and can be quite different. Three numerical experiments are carried out to examine the effects of distinguishing between the two rates. In the first experiment, mean survival times are calculated with differing rates. The second experiment involves harvesting with constant effort, and the third deals with harvesting at constant yield. It is found that equating birth rates with death rates can give overly optimistic results: survival rates too long, or catches too high. Population levels and yields are also calculated as a function of the correlation time in the fluctuating carrying capacity. Population levels and harvesting yields can be sensitive functions of fluctuation amplitude, correlation time and the ratio of reproduction ratio to death rates when simple variations of the logistic equation are used as harvesting models.

Continuation and optimization of the Born expansion in nonrelativistic quantum theory

Abstract Methods of von Neumann, Lanczos, and others, are applied to the non-relativistic scattering problem. It is shown that if the Born series is modified suitably, it will converge for almost any strength of interaction. Further, if suitable linear combinations of the modified Born approximations are used in the usual iteration process, the rate of convergence can be improved. The methods also apply to systems of integral equations which describe scattering of a particle from a complex system.

Selective synchrony of cells of differing cycle times.

Abstract If a growth inhibiting agent is applied periodically to a system of proliferating cells, travelling density waves in maturity space appear. These are synchrony waves; their group velocity depends on the average cycle time of the cells. If the cycle times of two types of cells differ significantly, only the type whose intermitotic period is close to the period at which the growth inhibitor is being applied comes into synchrony. A mathematical model is solved numerically to show the effect.

Mathematical description of holometabolous life cycles.

Abstract The deterministic continuous equations are developed for the population levels of the various life stages of holometabolous species. Special attention is paid to the larval stage in which the variable of weight, in addition to the traditional variables of age and time, is included; growth rate and the genetic variability of the growth rate are allowed for. A pair of equations is derived that permits computation of the larval population level without regard to the levels of the other life stages. Finite difference equations are developed, simple analytical forms for vital rates are adopted, and a numerical example is given.