On the absence of necessary conditions for linear evolution operators

There exist selfadjoint operators $A(t)(t \geqslant 0)$, whose resolvents depend smoothly on t, such that the initial value (Schrödinger) problem $du(t)/dt = iA(t)u(t),u(0) = f$ has a unique $({C^\infty })$ solution u for each f in a dense set in the underlying Hilbert space, with u depending continuously on f, and yet the intersection over $t \geqslant 0$ of the domain of $A(t)$ is {0}.