HONEYCOMB LATTICE POTENTIALS AND DIRAC POINTS

Abstract

In this article we study the spectral properties of the Schr ̈odinger operator HV = -Δ+V(x),xPR2, where the potential,V, is periodic and has honey-comb structure symmetry. For general periodic potentials the spectrum of HV, considered as an operator onL2(R2), is the union of closed intervals of continuous spectrum called the spectral bands. Associated with each spectral band are a band dispersion function,μ(k), and Floquet-Bloch states, u(x;k) = p(x;k)e(ik ̈x),where Hu(x;k) = μ(k)u(x;k) and p(x;k) is periodic with the periodicity of V(x).The quasi-momentum,k, varies over B, the first Brillouin zone [10]. Therefore, the time-dependent Schr ̈odinger equation has solutions of the form e(ipk ̈x -́μ(k)t)p(x;k).Furthermore, any finite energy solution of the initial value problem for the time-dependent Schr ̈odinger equation is a continuum weighted superposition, an integral dk, over such states. Thus, the time-dynamics are strongly influenced by the character of μ(k) on the spectral support of the initial data.