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

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

URL: http://arxiv.org/abs/2412.12011v1
Date: Mon, 16 Dec 2024 17:37:27 GMT
Title: Fermi's golden rule in tunneling models with quantum waveguides perturbed by Kato class measures
Authors: Sylwia Kondej, Kacper Ślipko,
Abstract summary: We show that the resolvent of the Hamiltonian has the second pole which reproduces the resonance at $z(rho)$ with the perturbations $z(rho)=mathcal E_beta ; n+mathcal O Big(frac exp(-sqrt2 |mathcal E_beta ;n| rho )rho Big)$ for $rho$ large and with the resonant energy $mathcal E_beta ;n$.
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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 $z(\rho)=\mathcal E_{\beta ; n}+\mathcal O \Big(\frac{ \exp(-\sqrt{2 |\mathcal E_{\beta ;n}| } \rho )}{\rho }\Big)$ 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.

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