Reviews in Mathematical Physics

We study the spectral properties of the two-dimensional Dirac operator on bounded domains together with the appropriate boundary conditions which provide a (continuous) model for graphene nanoribbons. These are of two types, namely, the so-called armchair and zigzag boundary conditions, depending on the line along which the material was cut. In the former case, we show that the spectrum behaves in what might be called a classical way; while in the latter, we prove the existence of a sequence of finite multiplicity eigenvalues converging to zero and which correspond to edge states.

In this paper, we are interested in a series of sub-hierarchies of the KP hierarchy introduced by Date et al. in [4], which we call the ℬ𝒞_r-KP hierarchy for r ∈ ℤ_≥0. In a unified way, we construct additional symmetries of the ℬ𝒞_r-KP hierarchy and show that all of them form a -algebra. Then, we introduce the constrained ℬ𝒞_r-KP hierarchy, and construct its additional symmetries and show that all of them form a Witt algebra.

In a semi-classical regime, we study a periodic magnetic Schrödinger operator in ℝ². This is inspired by recent experiments on artificial magnetism with ultra cold atoms in optical lattices, and by the new interest for the operator on the hexagonal lattice describing the behavior of an electron in a graphene sheet. We first review some results for the square (Harper), triangular and hexagonal lattices. Then, we study the case when the periodicity is given by the kagome lattice considered by Hou. Following the techniques introduced by Helffer–Sjöstrand and Carlsson, we reduce this problem to the study of a discrete operator on ℓ²(ℤ²;ℂ³) and a pseudo-differential operator on L²(ℝ;ℂ³), which keep the symmetries of the kagome lattice. We estimate the coefficients of these operators in the case of a weak constant magnetic field. Plotting the spectrum for rational values of the magnetic flux divided by 2πh where h is the semi-classical parameter, we obtain a picture similar to Hofstadter's butterfly. We study the properties of this picture and prove the symmetries of the spectrum and the existence of flat bands, which do not occur in the case of the three previous models.