Effective two level approximation

Bubak and Faifman performed the first comprehensive calculations for all combinations (except for the asymmetric spin flip) of muonic atom scattering in a broad energy range [15]. They used an improved form of the two level approximation of the Adiabatic Representation method in which the first two states are used in the expansion of the three-body wave function in Eq. 2.12 (i.e., N=2 in Eq. 2.16), hence Eq. 2.15 reduced to the system of two equations coupled with off-diagonal effective potentials U₁₂ and U₂₁. In order to overcome the shortcomings of the Adiabatic Representation method with a small N (see Section 2.1.3), a correction was made to the reduced masses, giving an appropriate asymptotic behavior of the solutions [98]. A considerable reduction of computational complexity of using only two levels allowed crude, but very useful, estimates of various cross sections. They quoted accuracies of typically 10-20% for most cases. It should be noted that the effective two level approximation, when applied to the muonic three body bound state problem, correctly predicts the existence of loosely bound states, which conventional two level approaches have failed to find.