Dynamical Lower Bounds for 1d Dirac Operators

with Dirichlet boundary conditions, acting on l2(N,C2), resp. L2([0,∞),C2), where c > 0 represents the speed of light, m ≥ 0 the mass of a particle, I2 is the 2× 2 identity matrix and V is a bounded real potential. In the discrete case D is the finite difference operator defined by (Dφ)(n) = φ(n+1)−φ(n), with adjoint (Dφ)(n) = φ(n − 1) − φ(n), and in the continuous case D = D = −i d dx . Model (1) in the continuous case is well known in relativistic quantum mechanics [1, 13], and the discrete version was introduced and studied in [6, 7]. The goal of this paper is to establish lower bounds on the dynamics associated to D(m, c) through the behaviour of the corresponding transfer matrices. To this end we will consider the time averaged q-th moments Aψ of the position operator