Hypercyclicity and mixing for cosine operator functions generated by second order partial differential operators

For Ω ⊂ R d open, we characterize when cosine operator functions generated by second order partial differential operators on L p ( Ω , μ ) and C 0 , ρ ( Ω ) , respectively, are hypercyclic and prove that this happens if and only if they are weakly mixing. In the case of d = 1 we give an easy to check characterization of when this happens. Moreover, mixing of these cosine operator functions is also characterized.