周期微扰量子系统中的伽莫夫矢量
Gamow Vectors in a Periodically Perturbed Quantum System
摘要：
我们分析了一维含时哈密顿量$H=-\partial_x^2\pm2δ(x)(1+2r\cosωt)$的波函数$ψ(x,t)$的行为，其中$ψ(x,0)$是紧支集的.我们发现，$ψ(x,t)$有一个Borel可和的扩展，包含许多条款的形式$\sum_{n=-\infty}^{\infty} e^{i^{3/2}\sqrt{-λ_{k}+n\omegai}|x|} A_{k,n} e^{-λ_{k}t+nωit}$，其中$λ_k$代表相关的共振。该表达式以严格和物理相关的方式定义了时间相关模型中所有频率和振幅的伽莫夫矢量和共振。对于小振幅（$|r|\ll 1$），一般初始条件下存在一个共振。我们计算了共振的位置，并讨论了它的物理意义有关的多光子电离。我们给出了定性的理论结果，以及在一般情况下的数值计算。
英文摘要：
We analyze the behavior of the wave function $ψ(x,t)$ for one dimensional time-dependent Hamiltonian $H=-\partial_x^2\pm2δ(x)(1+2r\cosωt)$ where $ψ(x,0)$ is compactly supported. We show that $ψ(x,t)$ has a Borel summable expansion containing finitely many terms of the form $\sum_{n=-\infty}^{\infty} e^{i^{3/2}\sqrt{-λ_{k}+n\omegai}|x|} A_{k,n} e^{-λ_{k}t+nωit}$, where $λ_k$ represents the associated resonance. This expression defines Gamow vectors and resonances in a rigorous and physically relevant way for all frequencies and amplitudes in a time-dependent model. For small amplitude ($|r|\ll 1$) there is one resonance for generic initial conditions. We calculate the position of the resonance and discuss its physical meaning as related to multiphoton ionization. We give qualitative theoretical results as well as numerical calculations in the general case.
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