 Operator conjugation by an exponential

For finite matrices, or formally on a common invariant domain where the expansion is meaningful, exponential conjugation is generated by the commutator map. Differentiating $F(s)=e^{sS}H{e}^{-sS}$ gives $F^{\prime }=[S,F]$, and its Taylor series yields the nested-commutator expansion. An anti-Hermitian $S$ makes this a unitary transformation. For perturbation theory this may be used order by order without asserting convergence for arbitrary unbounded operators.