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 $gasim O(1)$, 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.
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