05-148 Arne Jensen and Gheorghe Nenciu
The Fermi Golden Rule and its Form at Thresholds in Odd Dimensions (366K, pdf) Apr 26, 05
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Abstract. Let $H$ be a Schr\"{o}dinger operator on a Hilbert space $\cal H$, such that zero is a nondegenerate threshold eigenvalue of $H$ with eigenfunction $\Psi_0$. Let $W$ be a bounded selfadjoint operator satisfying $\langle\Psi_0,W\Psi_0\rangle>0$. Assume that the resolvent $(H-z)^{-1}$ has an asymptotic expansion around $z=0$ of the form typical for Schr\"{o}dinger operators on odd-dimensional spaces. Let $H(\varepsilon)=H+\varepsilon W$ for $\varepsilon>0$ and small. We show under some additional assumptions that the eigenvalue at zero becomes a resonance for $H(\varepsilon)$, in the time-dependent sense introduced by A. Orth. No analytic continuation is needed. We show that the imaginary part of the resonance has a dependence on $\varepsilon$ of the form $\varepsilon^{2+(\nu/2)}$ with the integer $\nu\geq-1$ and odd. This shows how the Fermi Golden Rule has to be modified in the case of perturbation of a threshold eigenvalue. We give a number of explicit examples, where we compute the ``location'' of the resonance to leading order in $\varepsilon$. We also give results, in the case where the eigenvalue is embedded in the continuum, sharpening the existing ones.

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