In this section I briefly review potential scattering following along the lines of ref. [4]. Let us consider for simplicity first the scattering of a spinless particle off a fixed potential. The wave function ф can then be expanded in partial waves

V(r) = £ u-NlY, m(i) (2.1)

Here, as everywhere vectors are bold faced and unit vectors carry addition­ally a hat. We use for the asymptotic scattering wave function ul(r) a linear combination of regular fi(r) and irregular gi(r) solutions of the free Hamil­tonian, so that all wave functions are real, thus simplifying the numerical calculations. The wavefunction ui is normalized to a ^-function in the energy by using the ansatz

Mr) = + ai 91 (r) + Ь"1 Xui (r)

Here M denotes the mass of the particle. The momentum k is related to the energy by E = h2k2/2M. For the variational principle ui (r) has to be regular at the origin, therefore gi(r) is the irregular solution gi regularized via

gi(r) = Ti(r)gi(r) with Ti(r) ^ r2l+l and Ti(r) 1 (2.3)

The regularisation factor Ti should approach 1 just outside the interaction region. A convenient choice is

where the limiting values are apparent in the different representations. A typical value for во is 1.1 fm-1. The calculation is rather insensitive to this parameter, see, however, the discussion below eq. (4.6).

The last term in eq. (2.2) accounts for the difference of the true solution of the scattering problem and the asymptotic form. Furthermore, in the region where Ti differs from this term one has to compensate the difference between gi(r) and gi(r). Since this term is different from zero only in a finite region, just somewhat larger than the interaction region, it can be well approximated by a finite number of square integrable terms. We will choose the terms in the form

Xvi(r) = ri+1e-e r (2.5)

where ви is an appropriately chosen set of parameters (see discussions in chapters 4.2 and 4.3).

Since fi and gl are not square integrable, we have to use Kohn’s variational principle [6] to determine the variational parameters ai and bvi via

where Hi denotes the Hamiltonian for the partial wave of angular momentum

l. It is easy to show [2], that all integrals in eq. (2.6) are well behaved if and only if the functions fi and gi are solutions of the free Hamiltonian to the energy E. See also the discussion in [5].