On the Stronger Statement of Levinson’s Theorem for the Dirac Equation

where the subscript l denotes the angular momentum. As is well know, there is degeneracy of states for the magnetic quantum number due to the spherical symmetry. Usually, this degeneracy is not explicitly expressed in the statement of Levinson’s theorem. In (1) the phase shifts are determined by comparing them with the phase shift at high energy in order to remove their indetermination of a multiple of 2π. The indetermination can also be removed by comparing them with the phase shifts of the free particle, that may be defined as zero so that δl(∞) in (1) can be removed. In fact, the experimental observation for Levinson’s theorem shows the jump of the phase shift at low energy in terms of changing the potential. In this letter we will use this convention for phase shifts. The second term with the sine square in (1) stands for the half bound state, that was first shown by Newton. From the Sturm-Liouville theorem, the number of nodes of the radial function at zero energy is equal to the number of the bound states for the Schrödinger equation with a non-singular spherically symmetric potential, so that it is related to the phase shift through Levinson’s theorem. Barthélémy first discuss Levinson’s theorem for the Dirac equation by the generalized Jost function. He stated that Levinson’s theorem is valid for positive and negative energies separately as in the non-relativistic case. But later this statement was found incorrect . The correct statement of Levinson’s theorem for the Dirac equation is :