08-113 Victor Dinu, Arne Jensen, and Gheorghe Nenciu

Non-exponential decay laws in perturbation theory of near threshold eigenvalues (380K, pdf) Jun 17, 08

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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.

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