On the numerical solution of the Hill–Wheeler equation

Abstract

A particular numerical method is discussed for solving the Hill–Wheeler equation, which is an integral equation that arises in the generator coordinate method analysis of problems in nuclear physics. The method is applicable to scattering problems with finite range potentials and exploits prior knowledge of the asymptotic form of the solution to convert the Hill–Wheeler equation into a Fredholm integral equation of the first kind. The resulting Fredholm equation is ill-posed, which often results in numerically unstable solutions. The method of regularization is used to produce numerically stable, approximate solutions. Optimizing the choice of regularization parameter is discussed and calculations are presented for an example problem.

Conserving symmetries in Bose-Einstein condensate dynamics requires many-body theory

“…A necessary condition for A to be conserved is that all moments d dt A n = 0 are time-independent, see Eq. (9). To what extent the observable A is conserved in the GP dynamics depends entirely on the extent to which the vectors A n Ψ are contained in the space (41).…”

“…A very popular method became known as the Hill-Wheeler integral equation, see e.g. [4,9]. An exhaustive list of its uses in nuclear and molecular physics is beyond the scope of this work.…”

Nucleon-nucleon interaction in the chromodielectric soliton model: Dynamics

“…This is a homogeneous Fredholm-type equation of the first kind, notoriously unstable numerically. Although some methods exist to make it stable (such as regularization [9], removal of the zero normalization eigenmodes [10], Gaussian transform [11], etc. ), we prefer to solve a differential equation approximately equivalent to the Hill-Wheeler equation, both for numerical stability and to facilitate comparison with analyses based on the Schrödinger equation.…”