Arne Jensen and Gheorghe Nenciu
The Fermi Golden Rule and its Form at Thresholds in Odd Dimensions
(366K, pdf)
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.