Relativistic Levinson theorem in two dimensions

作者：Shi‐Hai Dong, Xi-Wen Hou, Zhong-Qi Ma · 发表于：Physical Review A · 年份：1998 · DOI：10.1103/physreva.58.2160 · 被引用次数：66 · 研究领域：Quantum Mechanics and Non-Hermitian Physics、Quantum Mechanics and Applications、Quantum chaos and dynamical systems

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 ${\ensuremath{\eta}}_{j}(\ifmmode\pm\else\textpm\fi{}M)$ of the scattering states with the angular momentum $j$: ${\ensuremath{\eta}}_{j}(M)+{\ensuremath{\eta}}_{j}(\ensuremath{-}M)={\left({{(n}_{j}+1)\ensuremath{\pi},\mathrm{when}\mathrm{}\mathrm{a}\mathrm{}\mathrm{half}\mathrm{}\mathrm{bound}\mathrm{}\mathrm{state}\mathrm{}\mathrm{occurs}\mathrm{}\mathrm{at} E=M \mathrm{and} j=3/2 or \ensuremath{-}1/2}{{(n}_{j}+1)\ensuremath{\pi},\mathrm{when}\mathrm{}\mathrm{a}\mathrm{}\mathrm{half}\mathrm{}\mathrm{bound}\mathrm{}\mathrm{state}\mathrm{}\mathrm{occurs}\mathrm{}\mathrm{at} E=\ensuremath{-}M \mathrm{and} j=1/2 or \ensuremath{-}3/2}{{n}_{j}\ensuremath{\pi} ,\mathrm{the}\mathrm{}\mathrm{remaining}\mathrm{}\mathrm{cases}.}\right)$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.