Levinson's theorem for the Schrödinger equation in one dimension

Shi Hai Dong, Zhong Qi Ma

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

Levinson's theorem for the one-dimensional Schrödinger 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 Schrödinger 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 η+(0) [η-(0)] of the scattering states with the same parity at zero momentum as η+(0) + π/2 = n+π and η-(0) = n-π for the noncritical case, and η+(0) = n+π and η-(0) - π/2 = n-π for the critical case.
Original languageAmerican English
Pages (from-to)469-481
Number of pages420
JournalInternational Journal of Theoretical Physics
DOIs
StatePublished - 1 Jan 2000
Externally publishedYes

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Critical Case
Phase shift
One Dimension
Parity
Momentum
parity
theorems
Liouville theorem
Sturm's theorem
Scattering
Liouville's theorem
Sturm-Liouville
Zero
Phase Shift
Bound States
Theorem
infinity
momentum
phase shift
Odd

Cite this

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title = "Levinson's theorem for the Schr{\"o}dinger equation in one dimension",
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 η+(0) [η-(0)] of the scattering states with the same parity at zero momentum as η+(0) + π/2 = n+π and η-(0) = n-π for the noncritical case, and η+(0) = n+π and η-(0) - π/2 = n-π for the critical case.",
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Levinson's theorem for the Schrödinger equation in one dimension. / Dong, Shi Hai; Ma, Zhong Qi.

In: International Journal of Theoretical Physics, 01.01.2000, p. 469-481.

Research output: Contribution to journalArticle

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AB - Levinson's theorem for the one-dimensional Schrödinger 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 Schrödinger 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 η+(0) [η-(0)] of the scattering states with the same parity at zero momentum as η+(0) + π/2 = n+π and η-(0) = n-π for the noncritical case, and η+(0) = n+π and η-(0) - π/2 = n-π for the critical case.

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