  Recently, Ferrulli-Laptev-Safronov (2016arXiv161205304F) obtained eigenvalue
estimates for an operator associated to bilayer graphene in terms of $L^q$
norms of the (possibly non-selfadjoint) potential. They proved that for
$1<q<4/3$ all non-embedded eigenvalues lie near the edges of the spectrum of
the free operator. In this note we prove this for the larger range $1\leq q\leq
3/2$. The latter is optimal if embedded eigenvalues are also considered. We
prove similar estimates for a modified bilayer operator with so-called
"trigonal warping" term. Here, the range for $q$ is smaller since the Fermi
surface has less curvature. The main tool are new uniform resolvent estimates
that may be of independent interest and are collected in an appendix (in
greater generality than needed).
