  Using a recently introduced tensor network method, we study the density of
states of the lattice Schwinger model, a standard testbench for lattice gauge
theory numerical techniques, but also the object of recent experimental quantum
simulations. We identify regimes of parameters where the spectrum appears to be
symmetric and displays the expected continuum properties even for finite
lattice spacing and number of sites. However, we find that for moderate system
sizes and lattice spacing of $ga\sim O(1)$, the spectral density can exhibit
very different properties with a highly asymmetric form. We also explore how
the method can be exploited to extract thermodynamic quantities.
