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Given a Hilbert space, we investigate the well-posedness of the Cauchyproblem for the wave equation for operators with discrete non-negative spectrumacting on it. We consider the cases when the time-dependent propagation speedis regular, H\"older, and distributional. We also consider cases when it it isstrictly positive (strictly hyperbolic case) and when it is non-negative(weakly hyperbolic case). When the propagation speed is a distribution, weintroduce the notion of "very weak solutions" to the Cauchy problem. We showthat the Cauchy problem for the wave equation with the distributionalcoefficient has a unique "very weak solution" in appropriate sense, whichcoincides with classical or distributional solutions when the latter exist.Examples include the harmonic oscillator and the Landau Hamiltonian on $\mathbbR^n$, 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.

Wave equation for operators with discrete spectrum and irregular propagation speed