Victor Dinu, Arne Jensen, and Gheorghe Nenciu
Non-exponential decay laws in perturbation theory of near threshold 
eigenvalues
(380K, pdf)

ABSTRACT.  We consider a two channel model of the form 
$$ 
H_{\varepsilon}=\begin{bmatrix} 
H_{\rm op} & 0\\ 
0 & E_0 
\end{bmatrix} 
+\varepsilon\begin{bmatrix} 
0 & W_{12}\\ 
W_{21}&0 
\end{bmatrix} 
\quad 
\text{on} 
\quad 
\mathcal{H}=\mathcal{H}_{\rm op}\oplus \mathbf{C}. 
$$ 
The operator $H_{\rm op}$ 
is assumed to have the properties of a Schr\"{o}dinger operator in 
odd dimensions, with a threshold at zero. As the energy parameter $E_0$ is 
tuned past the threshold, we consider the survival probability 
$\lvert{\langle{\Psi_0},{e^{-itH_{\varepsilon}}\Psi_0}\rangle}\rvert^2, 
$ 
where $\Psi_0$ is the eigenfunction corresponding to eigenvalue $E_0$ for 
$\varepsilon=0$. We find non-exponential decay laws 
for $\varepsilon$ small and $E_0$ close to zero, 
provided that the resolvent of $H_{\rm op}$ is not at least Lipschitz continuous at the threshold zero.