We prove locality estimates, in the form of Lieb-Robinson bounds, for classical oscillator systems defined on a lattice. Our results hold for the harmonic system and a variety of anharmonic perturbations with long range interactions. The anharmonic estimates are applicable to a special class of observables, the Weyl functions, and the bounds which follow are not only independent of the volume but also the initial condition.

We prove Lieb-Robinson bounds for the dynamics of systems with an infinite dimensional Hilbert space and generated by unbounded Hamiltonians. In particular, we consider quantum harmonic and certain anharmonic lattice systems.