Eigenvalue estimates for bilayer graphene

© 2019, Springer Nature Switzerland AG. Recently, Ferrulli–Laptev–Safronov (2016) obtained eigenvalue estimates for an operator associated with bilayer graphene in terms of L q norms of the (possibly non-self-adjoint) potential. They proved that for 1 < q< 4 / 3 all nonembedded eigenvalues lie near the edges of the spectrum of the free operator. In this note, we prove this for the larger range 1 ≤ q≤ 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 is new uniform resolvent estimates that may be of independent interest and is collected in an appendix (in greater generality than needed).