\(L^p-L^q\) asymptotic behaviors of the solutions to the perturbed Schrödinger equations (Q1587241)

Let \(H= -\Delta+ V\) in \(L^2(\mathbb{R}^d)\), \(d\geq 3\), with a short range potential \(V\), such that the corresponding scattering system exists and is complete. Assume that \(H\) has no eigenvalue or resonance at zero. For that situation the asymptotics of the solution of the Schrödinger equation is studied as \(t\to\pm\infty\). It is given a sharp asymptotics in \(L^\infty(\mathbb{R}^d)\). In particular, the low energy part of \(e^{-itH}P_{ac}(H)\) is studied in detail.