Wave equation for operators with discrete spectrum and irregular propagation speed

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.