  We study two seminal approaches, developed by B. Simon and J. Kisy\'nski, to
the well-posedness of the Schr\"odinger equation with a time-dependent
Hamiltonian. In both cases the Hamiltonian is assumed to be semibounded from
below and to have constant form domain but a possibly non-constant operator
domain. The problem is addressed in the abstract setting, without assuming any
specific functional expression for the Hamiltonian. The connection between the
two approaches is the relation between sesquilinear forms and the bounded
linear operators representing them. We provide a characterization of continuity
and differentiability properties of form-valued and operator-valued functions
which enables an extensive comparison between the two approaches and their
technical assumptions.
