Half-bound states of a one-dimensional Dirac system: their effect on the Titchmarsh-Weyl M([lambda])-function and the scattering matrix

Abstract

We study the effect of the so-called half-hound states on the Titchmarsh-Weyl M(λ)· function and the S-matrix for a one dimensional Dirac system. For short range potentials with finite first (absolute) moments, we gave an M(λ) characterization of half bound states and, as a corollary, we deduce the behavior of the spectral function near the spectral gap endpoints. Further, we establish community of the S-matrix in momentum space and prove the Levinson theorem as a corollary to this analysis. We also obtain explicit asymptotics of the S-matrix for power-law potentials