Generalized parametrisation of a unitary matrix

The most general ansatz to construct a unitary transformation U = fiW) as an analytical function of an antihermitean operator W is a power series expansion, 

The coefficients have to satisfy a set of constraints such that U is unitary, i.e., 

Note that odd powers of W do not occur in this expansion because of the anti-hermiticity of W. With the requirement that different powers of W be linearly independent, we arrive at the following unitarity conditions for the coefficients  

The first coefficient oq is fixed apart from a global minus sign and can thus always be chosen as oq = 1. As it will be shown later, the even terms in the decoupled DK Hamiltonian do not depend on this choice for oq. Note that all constraints imposed on lower coefficients Oj, i — 0,2, 2k) have already been applied to express the condition for the next even coefficient 02 4-2 in Eqs. 

By using the general power series expansion for U all the infinitely many parametrisations of a unitary transformation are treated on equal footing. However, the question about the equivalence of these parametrisations for application in the Douglas-Kroll method, which represents a crucial point, is more subtle and will be analysed in the next section. It is especially not clear a priori, if the antihermitean matrix W can always be chosen in the appropriate way the mandatory properties of W, i.e., its oddness, antihermiticity and behaviour as a correct power in the external potential, have to be checked for every single transformation Ui of Eq. (73). 

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