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Analytical study of bound states in graphene nano-ribbons and carbon nanotubes: The variable phase method and the relativistic Levinson theorem

The problem of localized states in 1D systems with the relativistic spectrum, namely, graphene stripes and carbon nanotubes, has been analytically studied. The bound state as a superposition of two chiral states is completely described by their relative phase which is the foundation of the variable phase method (VPM) developed herein. Basing on our VPM, we formulate and prove the relativistic Levinson theorem. The problem of bound state can be reduced to the analysis of closed trajectories of some vector field. Remarkably, the Levinson theorem appears as the Poincare indices theorem for these closed trajectories. The reduction of the VPM equation to the non-relativistic and semi-classical limits has been done. The limit of the small momentum $p_y$ of the transverse quantization is applicable to arbitrary integrable potential. In this case the only confined mode is predicted.