  Given a Hilbert space, we investigate the well-posedness of the Cauchy
problem for the wave equation for operators with discrete non-negative spectrum
acting on it. We consider the cases when the time-dependent propagation speed
is regular, H\"older, and distributional. We also consider cases when it it is
strictly positive (strictly hyperbolic case) and when it is non-negative
(weakly hyperbolic case). When the propagation speed is a distribution, we
introduce the notion of "very weak solutions" to the Cauchy problem. We show
that the Cauchy problem for the wave equation with the distributional
coefficient has a unique "very weak solution" in appropriate sense, which
coincides with classical or distributional solutions when the latter exist.
Examples include the harmonic oscillator and the Landau Hamiltonian on $\mathbb
R^n$, uniformly elliptic operators of different orders on domains,
H\"ormander's sums of squares on compact Lie groups and compact manifolds,
operators on manifolds with boundary, and many others.
