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Full asymptotic expansion of transition probabilities in the adiabatic limit. (English) Zbl 0722.60086

Summary: We consider a two-level quantum mechanical system driven by an analytic time-dependent Hamiltonian of the form H(\(\epsilon\) t). In the adiabatic limit, \(\epsilon \ll 1\), the transition probability \({\mathcal P}(+,-)\) from one energy level (labelled by -) at time \(t=-\infty\) to the other (labelled by \(+)\) at time \(t=+\infty\) is known to behave as \({\mathcal P}(+,- )=\exp (-\alpha_{-1}\epsilon^{-1})\exp (\alpha_ 0)(1+O(\epsilon)).\) Using a simple iterative procedure generating Hamiltonians \(H_ 0=H,H_ 1,...,H_{N+1}\), we compute the full asymptotic expansion of the transition probability \[ \ln {\mathcal P}(+,-)=-\alpha_{-1}\epsilon^{- 1}+\alpha_ 0+\sum^{N}_{j=0}\alpha_ j\epsilon^ j+O(\epsilon^{N+1})\quad \forall N\geq 0. \]

MSC:

60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
81T27 Continuum limits in quantum field theory
35Q40 PDEs in connection with quantum mechanics
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