dorsal/arxiv
View SchemaLevinson's theorem for the Schr\"{o}dinger equation in two dimensions
| Authors | Shi-Hai dong, Xi-Wen Hou, Zhong-Qi Ma |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9806004 |
| URL | https://arxiv.org/abs/quant-ph/9806004 |
| DOI | 10.1103/PhysRevA.58.2790 |
Abstract
Levinson's theorem for the Schr\"{o}dinger equation with a cylindrically symmetric potential in two dimensions is re-established by the Sturm-Liouville theorem. The critical case, where the Schr\"{o}dinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison with Levinson's theorem in non-critical case, the half bound state for $P$ wave, in which the wave function for the zero-energy solution does not decay fast enough at infinity to be square integrable, will cause the phase shift of $P$ wave at zero energy to increase an additional $\pi$.
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"abstract": "Levinson\u0027s theorem for the Schr\\\"{o}dinger equation with a cylindrically\nsymmetric potential in two dimensions is re-established by the Sturm-Liouville\ntheorem. The critical case, where the Schr\\\"{o}dinger equation has a finite\nzero-energy solution, is analyzed in detail. It is shown that, in comparison\nwith Levinson\u0027s theorem in non-critical case, the half bound state for $P$\nwave, in which the wave function for the zero-energy solution does not decay\nfast enough at infinity to be square integrable, will cause the phase shift of\n$P$ wave at zero energy to increase an additional $\\pi$.",
"arxiv_id": "quant-ph/9806004",
"authors": [
"Shi-Hai dong",
"Xi-Wen Hou",
"Zhong-Qi Ma"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.58.2790",
"title": "Levinson\u0027s theorem for the Schr\\\"{o}dinger equation in two dimensions",
"url": "https://arxiv.org/abs/quant-ph/9806004"
},
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