Trapped modes in zigzag graphene nanoribbons

We study the scattering on an ultra-low potential in zigzag graphene nanoribbon. Using a mathematical framework based on the continuous Dirac model and the augmented scattering matrix, we derive a condition for the existence of a trapped mode. We consider the threshold energies where the continuous spectrum changes its multiplicity and show that the trapped modes may appear for energies slightly less than a threshold and that its multiplicity does not exceeds one. We prove that trapped modes do not appear outside the threshold, provided the potential is sufficiently small.