Vacuum charge: Another study in 1+1 dimensions

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Abstract
We use simple and direct methods to study questions related to charge fractionalization. The vacuum charge is obtained in a form reminiscent of the Levinson theorem relating the number of bound states to phase shifts. The potential ambiguity brought about by a ‘‘winding’’ of the chiral angle Δtheta is also studied. It is resolved by explicitly demonstrating the crossing pattern of the energy levels near E=0. This is achieved quite generally, as a result of applying the Sturm-Liouville theorem.