Fermi's golden rule in tunneling models with quantum waveguides perturbed by Kato class measures
In this paper we consider two dimensional quantum system with an infinite waveguide of the width $d$ and a transversally invariant profile. Furthermore, we assume that at a distant $ρ$ there is a perturbation defined by the Kato measure. We show that, under certain conditions, the resolvent of the Hamiltonian has the second sheet pole which reproduces the resonance at $z(ρ)$ with the asymptotics $z(ρ)=\mathcal E_{β; n}+\mathcal O \Big(\frac{ \exp(-\sqrt{2 |\mathcal E_{β;n}| } ρ)}{ρ}\Big)$ for $ρ$ large and with the resonant energy $\mathcal E_{β;n}$. Moreover, we show that the imaginary component of $z(ρ)$ satisfies Fermi's golden rule which we explicitly derive.
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