Levinson’s theorem for the Schrödinger equation in two dimensions

Shi Hai Dong, Xi Wen Hou, Zhong Qi Ma

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

Levinson’s theorem for the Schrödinger equation with a cylindrically symmetric potential in two dimensions is reestablished by the Sturm-Liouville theorem. The critical case, where the Schrödinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison to Levinson’s theorem in the noncritical case, the half bound state for the [Formula Presented] 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 the [Formula Presented] wave at zero energy to increase an additional [Formula Presented]. © 1998 The American Physical Society.
Original languageAmerican English
Pages (from-to)2790-2796
Number of pages2510
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
DOIs
StatePublished - 1 Jan 1998
Externally publishedYes

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theorems
Liouville theorem
infinity
energy
phase shift
wave functions
causes
decay

Cite this

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title = "Levinson’s theorem for the Schr{\"o}dinger equation in two dimensions",
abstract = "Levinson’s theorem for the Schr{\"o}dinger equation with a cylindrically symmetric potential in two dimensions is reestablished 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 to Levinson’s theorem in the noncritical case, the half bound state for the [Formula Presented] 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 the [Formula Presented] wave at zero energy to increase an additional [Formula Presented]. {\circledC} 1998 The American Physical Society.",
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Levinson’s theorem for the Schrödinger equation in two dimensions. / Dong, Shi Hai; Hou, Xi Wen; Ma, Zhong Qi.

In: Physical Review A - Atomic, Molecular, and Optical Physics, 01.01.1998, p. 2790-2796.

Research output: Contribution to journalArticle

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AB - Levinson’s theorem for the Schrödinger equation with a cylindrically symmetric potential in two dimensions is reestablished by the Sturm-Liouville theorem. The critical case, where the Schrödinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison to Levinson’s theorem in the noncritical case, the half bound state for the [Formula Presented] 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 the [Formula Presented] wave at zero energy to increase an additional [Formula Presented]. © 1998 The American Physical Society.

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