Relativistic Levinson theorem in two dimensions

In the light of the generalized Sturm-Liouville theorem, the Levinson theorem for the Dirac equation in two dimensions is established as a relation between the total number nⱼ of the bound states and the sum of the phase shifts ηⱼ(±M) of the scattering states with the angular momentum j: ${{η}}ⱼ(M)+{{η}}ⱼ({-}M)={({{(n}ⱼ+1){π},whenahalfboundstateoccursat E=M and j=3/2 or {-}1/2}{{(n}ⱼ+1){π},whenahalfboundstateoccursat E={-}M and j=1/2 or {-}3/2}{{n}ⱼ{π} ,theremainingcases.})$The critical case, where the Dirac equation has a finite zero-momentum solution, is analyzed in detail. A zero-momentum solution is called a half-bound state if its wave function is finite but does not decay fast enough at infinity to be square integrable.