Fermi's golden rule in tunneling models with quantum waveguides perturbed by Kato class measures

Abstract

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 $\rho$ 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 $z(\rho )$ with the asymptotics $z(\rho )={\mathcal{E}}_{\beta ;n}+\mathcal{O}\left(\frac{\exp (-\sqrt{2|{\mathcal{E}}_{\beta ;n}|}\rho )}{\rho }\right)$ for $\rho$ large and with the resonant energy ${\mathcal{E}}_{\beta ;n}$. Moreover, we show that the imaginary component of $z(\rho )$ satisfies Fermi's golden rule which we explicitly derive.

arXiv comment21 pages, 1 figure

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