dorsal/arxiv
View SchemaLevinson's theorem for the Schr\"{o}dinger equation in one dimension
| Authors | Shi-Hai Dong, Zhong-Qi Ma |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9903016 |
| URL | https://arxiv.org/abs/quant-ph/9903016 |
| DOI | 10.1007/s100530070079 |
| Journal | Int.J.Theor.Phys. 39 (2000) 469-481 |
Abstract
Levinson's theorem for the one-dimensional Schr\"{o}dinger equation with a symmetric potential, which decays at infinity faster than $x^{-2}$, is established by the Sturm-Liouville theorem. The critical case, where the Schr\"{o}dinger equation has a finite zero-energy solution, is also analyzed. It is demonstrated that the number of bound states with even (odd) parity $n_{+}$ ($n_{-}$) is related to the phase shift $\eta_{+}(0)[\eta_{-}(0)]$ of the scattering states with the same parity at zero momentum as $\eta_{+}(0)+\pi/2=n_{+}\pi, \eta_{-}(0)=n_{-}\pi$, for the non-critical case, $\eta_{+}(0)=n_{+}\pi, \eta_{-}(0)-\pi/2=n_{-}\pi$, for the critical case.
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"abstract": "Levinson\u0027s theorem for the one-dimensional Schr\\\"{o}dinger equation with a\nsymmetric potential, which decays at infinity faster than $x^{-2}$, is\nestablished by the Sturm-Liouville theorem. The critical case, where the\nSchr\\\"{o}dinger equation has a finite zero-energy solution, is also analyzed.\nIt is demonstrated that the number of bound states with even (odd) parity\n$n_{+}$ ($n_{-}$) is related to the phase shift $\\eta_{+}(0)[\\eta_{-}(0)]$ of\nthe scattering states with the same parity at zero momentum as\n$\\eta_{+}(0)+\\pi/2=n_{+}\\pi, \\eta_{-}(0)=n_{-}\\pi$, for the non-critical case,\n$\\eta_{+}(0)=n_{+}\\pi, \\eta_{-}(0)-\\pi/2=n_{-}\\pi$, for the critical case.",
"arxiv_id": "quant-ph/9903016",
"authors": [
"Shi-Hai Dong",
"Zhong-Qi Ma"
],
"categories": [
"quant-ph"
],
"doi": "10.1007/s100530070079",
"journal_ref": "Int.J.Theor.Phys. 39 (2000) 469-481",
"title": "Levinson\u0027s theorem for the Schr\\\"{o}dinger equation in one dimension",
"url": "https://arxiv.org/abs/quant-ph/9903016"
},
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