SPARSE 1D DISCRETE DIRAC OPERATORS I: QUANTUM TRANSPORT ROBERTO A. PRADO AND C ´ ESAR R. DE OLIVEIRA Abstract. Some dynamical lower bounds for one-dimensional discrete Dirac operators with different classes of sparse potentials are presented, and the particular role of the particle mass is emphasized. Keywords. Dirac operators, sparse potentials, quantum transport. 1. Introduction In this paper we consider discrete Dirac operators (1) D(m,c) := D 0 (m,c)+ V I 2 =  mc 2 cD ∗ cD −mc 2  + V I 2 , with boundary conditions so that (1) is self-adjoint, acting in ℓ 2 (N, C 2 ), where c> 0 represents the speed of light, m ≥ 0 the mass of the particle, I 2 is the 2 × 2 identity matrix and D is the finite difference operator defined by (Dϕ)(n)= ϕ(n + 1) − ϕ(n), with adjoint (D ∗ ϕ)(n)= ϕ(n − 1) − ϕ(n). The potential V : N → R is assumed to be polynomially bounded, that is, there exist constants a,b> 0 such that (2) |V (n)|≤ a(1 + n 2 ) b/2 , ∀n ∈ N = {1, 2, 3, ···}. Model (1) was introduced in [10, 11] as a relativistic version of the more common tight-binding Schr¨odinger operator, and it was further studied in [3, 18, 19]. The goal of the present paper is to present some lower bounds of the dynamics generated by D(m,c) with sparse potentials (see ahead for precise statements); due to the particular role played by the zero mass case (i.e., m = 0) in inducing quantum transport for (1), at least when the potential is given by i.i.d. random variables with a Bernoulli law [11], we will pay special attention to how the dynamical exponents depend on the mass m for sparse potentials. 1991 Mathematics Subject Classification. 81Q10. E-mail addresses: robertoprado@fct.unesp.br (RAP), oliveira@dm.ufscar.br (CRdeO). Fax: +55 18 3229-5385. RAP corresponding author, was supported by PROPe/UNESP and FUNDUNESP (Brazil). CRdeO was partially supported by CNPq (Brazil). 1