Fermi's golden rule in tunneling models with quantum waveguides perturbed by Kato class measures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909430658891776 |
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| author | Kondej, Sylwia Ślipko, Kacper |
| author_facet | Kondej, Sylwia Ślipko, Kacper |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12011 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fermi's golden rule in tunneling models with quantum waveguides perturbed by Kato class measures Kondej, Sylwia Ślipko, Kacper Mathematical Physics Quantum Physics 47B38, 81Q10, 81Q15, 81Q80 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. |
| title | Fermi's golden rule in tunneling models with quantum waveguides perturbed by Kato class measures |
| topic | Mathematical Physics Quantum Physics 47B38, 81Q10, 81Q15, 81Q80 |
| url | https://arxiv.org/abs/2412.12011 |