Geometrical structure of the space of all wavefunctions This is something I cannot grasp. The Schroedinger equation in 1D reads: If $\Psi (x,t) $ is at least differentiable in $t$ and at least two times differentiable in $x$, then
$$\frac{\partial\Psi (x,t)}{\partial t} = \frac{1}{i\hbar} H (x,t) \Psi(x,t) \tag{1}, $$ where $H$ is usually a linear function in $\frac{\partial^2}{\partial x^2}$, but otherwise can arbitrarily depend on $x$ and $t$. 
I have figured that if $H$ is totally independent of $t$, then the space of a "base solution" of (1) $\Psi (x,t)$ is the trivial fibering of $L^2 (\mathbb{R})$ by the unit circle, with the typical fiber a (class of equivalence of) function(s) in $L^2 (\mathbb{R})$, denote it by $\psi (x)$. But if the Hamiltonian has an arbitrary time-dependence, so that any separation of variables is impossible, then what geometrical structure would one give to the space of all solutions to the general (1)? 
 A: It might be a little surprising, but the space of solutions of the Schrödinger equation has the structure of a classical completely integrable infinite dimensional Hamiltonian system, please see for example the following article by Marmo and Vilasi.
In order to see that we can cast the Schrödinger equation in the form of Hamilton equations of motion as follows (from the article): 
By defining:
$$ q(r, t) = \mathrm{Re} \psi(r, t)$$
$$ p(r, t) = \mathrm{Im} \psi(r, t)$$
The (real and imaginary parts of the) Schrodinger equation can be written as:
$$ \frac{dq}{dt}= \frac{\delta \mathcal{H}}{\delta p}$$ 
$$ \frac{dp}{dt}= -\frac{\delta \mathcal{H}}{\delta q}$$ 
($\delta$ denotes the functional derivative), with the (classical) Hamiltonian of the system given by:
$$ \mathcal{H}[q, p] := \frac{1}{2\hbar}\int d^3r [\frac{1}{2m}((\nabla q)^2+(\nabla p)^2) + V(r)(q^2+ p^2)]$$
(It is assumed that the quantum Hamiltonian is of the form $H = -\frac{\hbar^2}{2m}\Delta+V(r)$).
Please see,  applications and generalization to nonlinear Schrödinger equations of this formalism in by Wu, Liu and Niu and Zhang and Wu.
Two remarks:
1) In the above Hamiltonian system, solutions related by a different global phase are distinct. The classical Hamiltonian is symmetric under a global phase change. In order to obtain the true space of solution we must perform a symplectic reduction of this symmetry. 
The resulting phase space is the space of true states (normalized wave functions modulo a global phase). It has the structure of an infinite dimensional complex projective space. This space is endowed with the Fubini-Study metric. The Schrödinger dynamics generate isometries of this space. In addition, according to Ashtekar and Schilling, the geodesic distances between two states provide the measurement probabilities.
2) The quantization of this infinite dimensional Hamiltonian system leads to a  second quantized version of the Schrödinger particles.
