Matrix product states, and operators, are powerful tools for the description of low energy eigenstates and thermal equilibrium states of quantum many-body systems in one spatial dimension. But in out-of-equilibrium scenarios, and for high energy eigenstates of generic systems, the scaling of entanglement with time and system size makes a direct application often impossible. However, MPS and more general TNS techniques can still be used to explore some of the most interesting dynamical properties.

We have recently introduced a method in which MPO techniques are combined with Chebyshev polynomial expansions to explore spectral properties of quantum many-body Hamiltonians. In particular, we show how this method can be used to probe thermalization of large spin chains without explicitly simulating their time evolution, as well as to compute full and local densities of states.