Abstract
I argue that, in the presence of a complete set of local integrals of motion, the whole spectrum of eigenvectors of a local Hamiltonian can be expressed efficiently in terms of a spectral tensor network. In D=1 dimensions, one can then use efficient tensor network manipulation techniques to compute a number of properties from the spectral MPS. These include the expectation value of any local observable for any energy eigenstate and (through perfect sampling) the estimation of time evolution, for arbitrarily long times, of any state that can be initially represented efficiently as a matrix product state.