Exactly Unitary Series Expansions

The most general ansatz to construct a unitary transformation U = /(W) as an analytic function of an antihermitean operator W is a power series expansion [611], 

An important observation is that odd powers of W do not occur in this expression because of the antihermiticity of W. With the requirement that different powers of W be linearly independent, the unitarity conditions for the coefficients are obtained. Their explicit form for the first few coefficients reads  

The first coefficient flg is fixed apart from a global minus sign and can therefore always be chosen as Aq = 1. All constraints imposed on lower coefficients (i = 0,2,. ..,2k) have already been applied to express the condition for the next even coefficient A2it+2 (H-60)-(11.65). For example, the depen- 

By using the general power series expansion for U all the infinitely many parametrizations of a unitary transformation are treated on an equal footing. However, the question about the equivalence of these parametrizations for application in decoupling Dirac-like one-electron operators needs to be studied. It is furthermore not clear a priori whether the antihermitean matrix W can always be chosen in the appropriate way the mandatory properties of W, i.e., its oddness, antihermiticity and behavior as a certain power in the chosen expansion parameter, have to be checked for every single transformation U applied to the untransformed or any pre-transformed Hamiltonian. Since the even expansion coefficients follow from the odd coefficients, the radius of convergence Rc of the power series depends strongly on the choice of the odd coefficients. 

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Evidently, this fomuila is not exact if fand vdo not connnute. However for short times it is a good approximation, as can be verified by comparing temis in Taylor series expansions of the middle and right-hand expressions in (A3,11,125). This approximation is intrinsically unitary, which means that scattering infomiation obtained from this calculation automatically conserves flux. 

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