The 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 $n_{j}$ of the bound states and the sum of the phase shifts $\eta_{j}(\pm M)$ of the scattering states with the angular momentum $j$: $$\eta_{j}(M)+\eta_{j}(-M)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$~~~=\left\{\begin{array}{ll} (n_{j}+1)\pi &{\rm when~a~half~bound~state~occurs~at}~E=M ~~{\rm and}~~ j=3/2~{\rm or}~-1/2\\ (n_{j}+1)\pi &{\rm when~a~half~bound~state~occurs~at}~E=-M~~{\rm and}~~ j=1/2~{\rm or}~-3/2\\ n_{j}\pi~&{\rm the~rest~cases} . \end{array} \right.$$ \noindent 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.