  We study a non-autonomous, non-linear evolution equation on the space of
operators on a complex Hilbert space. We specify assumptions that ensure the
global existence of its solutions and allow us to derive its asymptotics at
temporal infinity. We demonstrate that these assumptions are optimal in a
suitable sense and more general than those used before. The evolution equation
derives from the Brocket-Wegner flow that was proposed to diagonalize matrices
and operators by a strongly continuous unitary flow. In fact, the solution of
the non-linear flow equation leads to a diagonalization of Hamiltonian
operators in boson quantum field theory which are quadratic in the field.
