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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 *U*_{12} and *U*_{21}. 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.

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