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 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)=E_β ; n+O ( exp(-√2 |E_β ;n| ρ )/ρ ) for ρ large and with the resonant energy E_β ;n. Moreover, we show that the imaginary component of z(ρ) satisfies Fermi's golden rule which we explicitly derive.