Abstract
We show that the existence of the family of self-adjoint Lyapunov operators introduced in Strauss (J. Math. Phys. 51:022104, 2010) allows for the decomposition of the state of a quantum mechanical system into two parts: A backward asymptotic component, which is asymptotic to the state of the system in the limit t→-∞ and vanishes at t→∞, and a forward asymptotic component, which is asymptotic to the state of the system in the limit t→∞ and vanishes at t→-∞. We demonstrate the usefulness of this decomposition for the description of resonance phenomena by considering the resonance scattering of a particle off a square barrier potential. We show that the evolution of the backward asymptotic component captures the behavior of the resonance. In particular, it provides a spatial probability distribution for the resonance and exhibits its typical decay law.
Original language | English |
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Pages (from-to) | 2179-2190 |
Number of pages | 12 |
Journal | International Journal of Theoretical Physics |
Volume | 50 |
Issue number | 7 |
DOIs | |
State | Published - Jul 2011 |
Bibliographical note
Funding Information:Acknowledgements Y. Strauss acknowledges support from the Israeli Science Foundation (Grants No. 1169/06 and 1105/10). J. Silman and S. Machnes acknowledge support from the Israeli Science Foundation (Grant No. 784/06). J. Silman also acknowledges the support of the Inter-University Attraction Poles Programme (Belgian Science Policy) under Project IAP-P6/10 (Photonics@be) and of the FNRS.
Funding
Acknowledgements Y. Strauss acknowledges support from the Israeli Science Foundation (Grants No. 1169/06 and 1105/10). J. Silman and S. Machnes acknowledge support from the Israeli Science Foundation (Grant No. 784/06). J. Silman also acknowledges the support of the Inter-University Attraction Poles Programme (Belgian Science Policy) under Project IAP-P6/10 (Photonics@be) and of the FNRS.
Funders | Funder number |
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Israeli Science Foundation | 1169/06, 1105/10, 784/06, IAP-P6/10 |
Fonds De La Recherche Scientifique - FNRS |
Keywords
- Lyapunov operator
- Resonance
- Semigroup decomposition
- Transition decomposition