Exact effective Hamiltonian theory. II. Polynomial expansion of matrix functions and entangled unitary exponential...

Our recent exact effective Hamiltonian theory (EEHT) for exact analysis of nuclear magnetic resonance (NMR) experiments relied on a novel entanglement of unitary exponential operators via finite expansion of the logarithmic mapping function. In the present study, we introduce simple alternant quotient expressions for the coefficients of the polynomial matrix expansion of these entangled operators. These expressions facilitate an extension of our previous closed solution to the Baker-Campbell-Hausdorff problem for SU(N) systems from N< or =4 to any N, and thereby the potential application of EEHT to more complex NMR spin systems. Similarity matrix transformations of the EEHT expansion are used to develop alternant quotient expressions, which are fully general and prove useful for evaluation of any smooth matrix function. The general applicability of these expressions is demonstrated by several examples with relevance for NMR spectroscopy. The specific form of the alternant quotients is also used to demonstrate the fundamentally important equivalence of Sylvester's theorem (also known as the spectral theorem) and the EEHT expansion.