Inequivalence between the spectral functions of local vs translationally invariant observables

Theoretical predictions for local observables are often used interchangeably with those of their translationally invariant counterparts. We argue that while this is justified for the equal-time expectation values of observables, it cannot be done in the context of dynamical correlation functions. We specifically address this question in the context of the eigenstate thermalization hypothesis (ETH). We show that for the nonintegrable spin-1 tilted-field Ising chain the spectral functions of local and translationally invariant observables are different in energy eigenstates at "infinite temperature". We discuss the origin of those differences both in systems with open and periodic boundary conditions.