dorsal/arxiv
View SchemaThe Relativistic Levinson Theorem in Two Dimensions
| Authors | Shi-Hai dong, Xi-Wen Hou, Zhong-Qi Ma |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9806006 |
| URL | https://arxiv.org/abs/quant-ph/9806006 |
| DOI | 10.1103/PhysRevA.58.2160 |
Abstract
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.
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"abstract": "In the light of the generalized Sturm-Liouville theorem, the Levinson theorem\nfor the Dirac equation in two dimensions is established as a relation between\nthe total number $n_{j}$ of the bound states and the sum of the phase shifts\n$\\eta_{j}(\\pm M)$ of the scattering states with the angular momentum $j$:\n$$\\eta_{j}(M)+\\eta_{j}(-M)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$\n$$~~~=\\left\\{\\begin{array}{ll} (n_{j}+1)\\pi \u0026{\\rm\nwhen~a~half~bound~state~occurs~at}~E=M ~~{\\rm and}~~ j=3/2~{\\rm or}~-1/2\\\\\n(n_{j}+1)\\pi \u0026{\\rm when~a~half~bound~state~occurs~at}~E=-M~~{\\rm and}~~\nj=1/2~{\\rm or}~-3/2\\\\ n_{j}\\pi~\u0026{\\rm the~rest~cases} . \\end{array} \\right. $$\n \\noindent The critical case, where the Dirac equation has a finite\nzero-momentum solution, is analyzed in detail. A zero-momentum solution is\ncalled a half bound state if its wave function is finite but does not decay\nfast enough at infinity to be square integrable.",
"arxiv_id": "quant-ph/9806006",
"authors": [
"Shi-Hai dong",
"Xi-Wen Hou",
"Zhong-Qi Ma"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.58.2160",
"title": "The Relativistic Levinson Theorem in Two Dimensions",
"url": "https://arxiv.org/abs/quant-ph/9806006"
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