What are range and domain of Hamiltonian?
Hamiltonian is not a function, but an operator (a function in a broader sense, if you'd like):
$${\hat {H}}=-{\frac {\hbar ^2}{2m}}\nabla ^2+V(\bar{r},t)$$
It is applied to some function (our supposed $\psi$) and yields some other function. So its domain is the set of all doubly differentiable functions on our underlying space (the space itself can be 1D, 3D, or whatever else, depending on the nature of the problem), and the range is even broader than that, because the derivatives (or $V$, for that matter) do not have to be differentiable.