| Contents |
| Let $k$ be a resonance (scattering frequency) for the operator $H =-\Delta \,+ \,V(x)$. This means that $\Im k$ is negative and
the problem
$$
(-\Delta \,+ \,V(x)\,) \varphi \,= k^2 \varphi,
$$
has a nontrivial outgoing solution. Since the outgoing condition rules out square integrability, it is reasonable to truncate
the resonant solution $\varphi$ in a compact region. One then expects that the evolution of the truncated solution under the Schrodinger
equation, will exhibit an exponential behavior, when the time $t$ approaches infinity.
We shall review a method, due to R. Lavine, that proves this fact on the half line $[0,\infty)$ and then extend it to systems of Schr\"odinger equations,
$$-\Delta u + V(x)u + g(x)v \,=k^2 u \, ,
$$
$$
-\Delta v + W(x)v + g(x)u \, =k^2 v \, .
$$ |
|