  We study the energy spectrum of tight-binding Hamiltonian for regular
hyperbolic tilings. More specifically, we compute the density of states using
the continued-fraction expansion of the Green function on finite-size systems
with more than $10^9$ sites and open boundary conditions. The coefficients of
this expansion are found to quickly converge so that the thermodynamical limit
can be inferred quite accurately. This density of states is in stark contrast
with the prediction stemming from the recently proposed hyperbolic band theory.
Thus, we conclude that the fraction of the energy spectrum described by the
hyperbolic Bloch-like wave eigenfunctions vanishes in the thermodynamical
limit.
