TY - JOUR
T1 - Transition Decomposition of Quantum Mechanical Evolution
AU - Strauss, Y.
AU - Silman, J.
AU - Machnes, S.
AU - Horwitz, L. P.
N1 - 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.
PY - 2011/7
Y1 - 2011/7
N2 - 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.
AB - 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.
KW - Lyapunov operator
KW - Resonance
KW - Semigroup decomposition
KW - Transition decomposition
UR - http://www.scopus.com/inward/record.url?scp=79958087803&partnerID=8YFLogxK
U2 - 10.1007/s10773-011-0689-y
DO - 10.1007/s10773-011-0689-y
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AN - SCOPUS:79958087803
SN - 0020-7748
VL - 50
SP - 2179
EP - 2190
JO - International Journal of Theoretical Physics
JF - International Journal of Theoretical Physics
IS - 7
ER -