  Energy filter methods in combination with quantum simulation can efficiently
access the properties of quantum many-body systems at finite energy densities
[Lu et al. PRX Quantum 2, 020321 (2021)]. Classically simulating this algorithm
with tensor networks can be used to investigate the microcanonical properties
of large spin chains, as recently shown in [Yang et al. Phys. Rev. B 106,
024307 (2022)]. Here we extend this strategy to explore the properties of
off-diagonal matrix elements of observables in the energy eigenbasis,
fundamentally connected to the thermalization behavior and the eigenstate
thermalization hypothesis. We test the method on integrable and non-integrable
spin chains of up to 60 sites, much larger than accessible with exact
diagonalization. Our results allow us to explore the scaling of the
off-diagonal functions with the size and energy difference, and to establish
quantitative differences between integrable and non-integrable cases.
