Eigenvalue estimates for bilayer graphene

TL;DR

This paper extends eigenvalue estimates for bilayer graphene operators to a larger range of $L^q$ norms, introduces estimates for a modified operator with trigonal warping, and develops new resolvent bounds of broader applicability.

Contribution

It broadens the eigenvalue estimate range for bilayer graphene operators and introduces resolvent estimates that could benefit related spectral analysis.

Findings

Eigenvalue estimates now valid for $1 \,\leq\, q \leq 3/2$

Established estimates for operators with trigonal warping

Developed new uniform resolvent estimates of independent interest

Abstract

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).