Levinson theorem for the Dirac equation in the presence of solitons in (1+1) dimensions

We study the Levinson theorem in (1+1) dimensions for a Dirac particle coupled to a background pseudoscalar field, which is allowed to evolve into a soliton with arbitrary topological charge. We demonstrate the difficulty with defining the phase shifts when solitons are present, and then present a prescription for defining the phase shifts when the background field has arbitrary boundary values. We find that a modified form of the Levinson theorem holds, in which half the total number of threshold bound states at zero strength of the potential needs to be subtracted from the total number of bound states. We then present the Levinson theorem in a form that is universal to both relativistic and nonrelativistic quantum mechanics, which also includes all the exceptional cases. We also find generalizations to the Levinson theorem by finding relations between the bound states and the values of the phase shifts at each boundary of the continua separately. We relate the values of the phase shifts at the boundaries of a given continuum to the spectral deficiency in that continuum. We also relate the values of phase shifts at infinite energies to the topological charge of the solitons and their adiabatic contribution to the vacuum polarization that they induce.