Non-separable solutions of the Schroedinger equation

This is indeed possible only in some situations, e.g. when the continuous spectrum is absent (it may also consist of a single point, see Valter Moretti's comment below). A sufficient condition for that to be true is that either the Hamiltonian is compact or it has compact resolvent.

Sadly, very few interesting Hamiltonians satisfy that property (an example being the harmonic oscillator). In general, the solution of the Schrödinger equation exists for any initial condition $\psi_0\in\mathscr{H}$ (the Hilbert space), using the unitary one-parameter strongly continuous group $e^{-it H}$ associated to the self-adjoint Hamiltonian $H$ by Stone's theorem. The solution at any time $t\in \mathbb{R}$ is then simply written $$\psi(t)=e^{-itH}\psi_0\; .$$ Such solution is continuous in time, and with respect to initial data, but it is differentiable in time only if $\psi_0\in D(H)$, where $D(H)$ is the domain of the self-adjoint operator $H$. In the language of analysis of PDEs, that means that the Schrödinger equations, for self-adjoint Hamiltonians, are globally well posed on $\mathscr{H}$.

Essentially, separation of variables in the time-independent Schroedinger equation amounts to diagonalizing the Hamiltonian. One can see this easily by considering the case where the Hilbert space is finite-dimensional, and the Hamiltonian is a Hermitian matrix.

In case of a partially continuous spectrum one gets the same, except that the sum must be replaced by an integral over a set of labels that resolves the complete spectrum. The continuous spectrum is labelled by the momenta of the possible scattering states. The transformation to the diagonalized operator is given by the so-called Moeller operator.