Finite Dimensional Function Space Research Articles

Investigation of the effective interaction by means of a modified shell model

The effective InteractionU for a shell model with hamiltonianHm is investigated by means of a “mixed” model with hamiltonianHm. For this model the Bornv. Neumann expansion of the effective interactionŨ reduces to its first two terms. Within the finite dimensional function space ℜn used in the shell model calculation the operatorsU andŨ are exactly equivalent. This is used to show that an energy independent approximation forU- if it is possible at all-is necessarily equal to the residual interaction, i.e. to the difference between nucleon-nucleon interaction and shell model interaction.

Projective solution of integral equations arising in electric and magnetic field problems

Abstract Many integral equations arising in electric and magnetic field problems have Green's function kernels, which cause difficulty in integration because of their essential singularities. By performing a simple coordinate rotation; these singularities can be shifted entirely onto one coordinate axis, so that iterated weighted numerical quadratures may be applied to evaluate exactly the projective representation of such integral operators on suitable finite-dimensional function spaces. Standard projective methods may therefore be applied to solve such integral equations accurately with very little computational effort.

On semi-parabolic Riemann surfaces

On finite surfaces, the class of harmonic functions which are constant on each contour is a finite-dimensional vector space of functions with finite Dirichlet norm. This paper considers the corresponding class of functions on bordered surfaces of class SOg and generalizes some of the properties of harmonic measures on finite surfaces. In particular, for generalized harmonic measures, we investigate the level curves and their orthogonal trajectories. The principal results, Theorems 4.1 and 4.4, state, in a sense made precise, that almost all of the level curves of a generalized harmonic measure are analytic Jordan curves and almost all of their orthogonal trajectories begin and end on the border given in the definition of the surface. These results have application to the level curves of a Green's function via a theorem of Kuramochi. We also consider the question on a parabolic surface as to when a harmonic differential with finite norm and integral periods is a weak limit of period reproducing differentials.