Transforming a perturbed Hamiltonian H=H0+V to an effective Heff. Theory and Expansions

What is it about?

(Uploaded: 2018-04-18). This should be read in continuation of the section "What is it about" in my Kudos-description of "A projector formulation of the Van Vleck Transformation (VVT); Part I. Degenerate case". In that paper we transferred to VVT the projection operator formalism laid out earlier by Bloch whose very different approach builds on simple geometry and leads to very simple expansions. His Heff is, however, non-hermitian while it becomes hermitian with use successive VVT: W =...exp(iS2) exp(iS1) where S1, S2, ... are hermitian. Furthermore: It can sometimes be important that the exponential operators and the Hausdorff-formula makes it possible to write Heff in terms of multiple commutators. That was for instance crucial when we considered the renowned Foldy-Wouthoysen Transformation (FWT) as a particularly simple special case of VVT. At the time of the present paper a number of seemingly different approaches for a hermitian Heff had been suggested. I now formulate a theorem of uniqueness and some recurrence relations which prove that most transformations end up as being identical to the one I to refer to Soliverez [] who found his by modifying Bloch's geometric approach. My theorem of uniqueness, which is extremely simple to use, reveals that it is the same as w = exp(is) where sn in s= s1+ s2+... is chosen to be hermitian and odd. I was not aware that Klein previously had identified this same w with a transformation due to des Cloizeaux who, prior to Soliverez, had presented a similar adjustment of Bloch's approach. Klein proved the identity by a theorem of uniqueness formulated by himself. Intuitively this is much more appealing than mine - but it is also rather more difficult to use. And actually Klein's proof suffers from a crucial error. None the less his result is correct. See the Kudos-description of my own later brief article: Some comments on Klein's review "Degenerate perturbation theory" (Mol. Pys. 1974, 27,p. 33) Klein called w the canonical version of VVT - and that name has remained.

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Why is it important?

The most important results of the present article must be the following two which are closely re-lated. I: Recurrence relations which practically any approach must be in accordance with. II: The simple theorem of uniqueness for the canonical VVT. This latter theorem is extremely simple to use and it seems to have given the first correct proof that the exponential transformation w is identical to the non-exponential transformations due to des Cloizeaux and to Soliverez. These can therefore be carried out by use of the Hausdorff formula with its multiple commutators - important in connection with Lie algebra and linked cluster expansions. The recurrence relations are particularly simple in this canonical case. I have used them to find a recurrence relation for the difference between the canonical Heff and the hermitian part of Bloch's simple fundamental Heff. It turns out that this difference starts in order 5 in the degenerate case and order 4 in the near degenerate. The recurrence relations work so well together with the Hausdorff formula that, in case of Dirac's relativistic Hamiltonian for the electron, I have been able to calculate Heff through the order 6 with the canonical version w of VVT as well as with the successive version W used originally by Foldy and Wouthoysen to reach the order 6. The two Heff do not differ in order 4 but they do in order 6. Odd orders vanish.

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