Global minimization in few-body systems

Optimization techniques have been extensively used in a variety of physical problems in Many-Body systems such as the construction of optical potentials, phase shift analyses, variational calculations, neural network training, etc. In the field of Few-Body systems however, their use is rather limited although there are cases where the method could be very useful. A subtle feature is that when the original problem is transformed to an optimization one, the resulting objective function possesses a host of local minima. From these minima only a few may correspond to physically acceptable solutions, hence local optimization is inadequate and GO methods are required. A typical example is the evaluation of the spectrum of a heavy three-molecular system using Faddeev-type methods where one has to evaluate the plethora of low lying states Eν , ν = 1, 2, · · · , N in a relatively small energy window ΔE. The spectrum is usually obtained using standard few-body methods based on finite element techniques the accuracy of which depends on the polynomial chosen, the distribution and density of the relevant grid points employed, and the handling of the resulting huge matrices. The calculations are often tedious and in certain cases problematic as the calculated spectrum depends on the total accuracy of solving the three-body equations and each individual evaluation of a spectral point, results in a binding energy that lies in an energy range where several other binding energies exist and therefore it is not readily clear for which ν the binding energy found corresponds. The problem becomes even more complex when one wants to construct or improve three-molecular forces under the constrain that these should reproduce the lowlying spectrum of the system. These forces should complement the highly realistic and well known interatomic interactions and a fine tuning to evaluate the parameters of the three-body force so that the low lying states are simultaneously reproduced is desirable. In such cases the use of a GO is necessary. Yet another characteristic example is the location of the zeros of the Jost function Nuclear Physics A 790 (2007) 655c–658c