Eigenvalue Estimates for Bilayer Graphene

Abstract

Recently, 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 ≤ 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 are new uniform resolvent estimates that may be of independent interest and are collected in an appendix (in greater generality than needed).

Weak coupling limit for Schrödinger-type operators with degenerate kinetic energy for a large class of potentials

“…The validity of the Tomas-Stein theorem crucially depends on the curvature of the underlying manifold. A slight modification of our proof (see, e.g., [7,8]) shows that the result of Theorem 1.1 continues to hold for general Schrödinger-type operators (with a suitable modification of the local regularity assumption) of the form (1.1) as long as the Fermi surface S = {ξ ∈ R d : T (ξ ) = 0} is smooth and has everywhere non-vanishing Gaussian curvature. For example, if T is elliptic at infinity of order 2d/(d +1) ≤ s < d, then the assumption on the potential becomes V ∈ [20,Theorem 2.1].…”

“…The moment-type condition on the potential in that theorem is unnecessary, regardless of whether the kinetic energy is radial or not. A straightforward generalization to the case where S has at least k non-vanishing principal curvatures can be obtained from the results of [8,17]. In that case the global decay assumption has to be strengthened to V ∈ k+2 2 L d s .…”

Limiting absorption principle on $L^p$-spaces and scattering theory

“…In this paper, for a large class of operators T (D) on X d , we study uniform resolvent estimates, Hölder continuity of the resolvent and Carleman type inequalities for Fourier multipliers on X d , where X = R or X = Z. The uniform resolvent estimates for a Fourier multipliers are investigated in [4] and [5] in the duality line when X = R in order to study the Lieb-Thirring type bounds for fractional Schrödinger operators and Dirac operators. One of the purpose is to prove the uniform resolvent estimates away form the duality line and to extend to the case of X = Z.…”

“…Example 1. Suppose that M λ ∩ supp χ has at least m nonvanishing principal curvature curvature at every point, then (5) holds for k = m/2 by the stationary phase theorem.…”

Lieb-Thirring inequalities for an effective Hamiltonian of bilayer graphene

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Combining the methods of Cuenin [7] and Borichev-Golinskii-Kupin [4], [5], we obtain the so-called Lieb-Thirring inequalities for nonselfadjoint perturbations of an effective Hamiltonian for bilayer graphene.

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