  We study convergence rates of the Trotter-Kato splitting $e^{A+L} = \lim_{n
\to \infty} (e^{L/n} e^{A/n})^n$ in the strong operator topology. In the first
part, we use complex interpolation theory to treat generators $L$ and $A$ of
contraction semigroups on Banach spaces, with $L$ relatively $A$-bounded. In
the second part, we study unitary dynamics on Hilbert spaces and develop a new
technique based on the concept of energy constraints. Our results provide a
complete picture of the convergence rates for the Trotter splitting for all
common types of Schr\"odinger and Dirac operators, including singular,
confining and magnetic vector potentials, as well as molecular many-body
Hamiltonians in dimension $d=3$. Using the Brezis-Mironescu inequality, we
derive convergence rates for the Schr\"odinger operator with $V(x)=\pm
|x|^{-a}$ potential. In each case, our conditions are fully explicit.
