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We study two seminal approaches, developed by B. Simon and J. Kisy\'nski, tothe well-posedness of the Schr\"odinger equation with a time-dependentHamiltonian. In both cases the Hamiltonian is assumed to be semibounded frombelow and to have constant form domain but a possibly non-constant operatordomain. The problem is addressed in the abstract setting, without assuming anyspecific functional expression for the Hamiltonian. The connection between thetwo approaches is the relation between sesquilinear forms and the boundedlinear operators representing them. We provide a characterization of continuityand differentiability properties of form-valued and operator-valued functionswhich enables an extensive comparison between the two approaches and theirtechnical assumptions.

On the Schr\"odinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain