MaPhySto
The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

MPS-RR 2004-31
December 2004

# The Fermi Golden Rule at Thresholds

by: Arne Jensen, Gheorghe Nenciu

## Abstract

Let $H$ be a Schrödinger operator on a Hilbert space $\mathcal{H}$, 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 Schrödinger 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 $\varepsilon$.

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