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

TL;DR

This paper analyzes quantum waveguides with distant perturbations described by Kato measures, deriving resonance behavior and explicitly confirming Fermi's golden rule for the imaginary part of the resonance energy.

Contribution

It provides a rigorous analysis of resonances and Fermi's golden rule in quantum waveguides perturbed by Kato class measures, including explicit asymptotics.

Findings

Resonance poles are shown to exist on the second sheet of the resolvent.

Resonance energies asymptotically approach the unperturbed energies with exponential decay.

The imaginary part of the resonance energy satisfies Fermi's golden rule.

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.