Transition Decomposition of Quantum Mechanical Evolution
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.
KeywordsLyapunov operator Transition decomposition Resonance Semigroup decomposition
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