Infinite-Order Quasi-Degenerate DPT

Alternatively to the hermitean effective Hamiltonian just discussed, we can consider a nonhermitean effective Hamiltonian L obtained by a projecting similarity transformation in intermediate normalization. Actually the first formulation of quasidegenerate DPT by Rutkowski and Schwarz [76] was given in terms of a nonhermitean effective Hamiltonian. This is particularly useful if one wants to extend quasidegenerate DPT to infinite order. 

It is possible to transform the nonhermitean L to a hermitean L by means of the transformation 

To construct first L and then transform to L in the indicated way is rather involved in the frame of perturbation theory. There the direct way given in the preceding section is easier and more transparent. However, the approach via L looks preferable in a non-expanded iterative context. 

One starts with X = Xq and w = 1, from which one constructs then AL of the iteration cycle. In the first iteration cycle one has 

One then proceeds until self-consistency. Actually one will perform this iteration in terms of the matrix representation of L and w. After one has constructed L in intermediate normalization, one can hermitize it by the non-similarity transformation (441). 

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