Remarks on the space-time behavior of scattering solutions to the Schrödinger equations (Q1102458)

The author considers the space-time behavior of the Schrödinger evolution \(e^{itH}\phi\) with \(\phi\) in the continuous spectral subspace of the self-adjoint operator H in \(L^ 2({\mathbb{R}}^ n)\). One of the main results is the following: Every scattering solution which has the estimate \(| (e^{-itH}\phi)(x)| \leq C| t|^{- \alpha}(1+| x|)^{-\beta}\), \(| t| \geq 1\), \(x\in {\mathbb{R}}^ n\), for some \(\alpha\),\(\beta\in {\mathbb{R}}\) with \(\alpha +\beta >n/2\), vanishes identically. This shows that the well-known estimate which corresponds to the case \(\alpha +\beta =n/2\) is optimal.