BooksStandalone TitlesCollected Volumepp. 517–563

The LpL^{p}-boundedness of wave operators for four-dimensional Schrödinger operators

  • Kenji Yajima

    Gakushuin University, Tokyo, Japan
The $L^{p}$-boundedness of wave operators for four-dimensional Schrödinger operators cover
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Abstract

We prove that the low energy parts of the wave operators W±W_{\pm} for Schrödinger operators H=Δ+V(x)H=-\Delta+V(x) on R4\mathbb{R}^{4} are bounded in Lp(R4)L^{p}(\mathbb{R}^{4}) for 1<p21<p\leq 2 and are unbounded for 2<p2<p\leq\infty if HH has resonances at the threshold. If HH has eigenfunctions only at the threshold, it has recently been proved that they are bounded in Lp(R4)L^{p}(\mathbb{R}^{4}) for 1p<41\leq p<4 in general and for 1p<1\leq p<\infty if all threshold eigenfunctions φ\varphi satisfy R4xjV(x)φ(x)dx=0\int_{\mathbb{R}^{4}}x_{j}V(x)\varphi(x)\,dx=0 for 1j41\leq j\leq 4. We prove in this case that they are unbounded in Lp(R4)L^{p}(\mathbb{R}^{4}) for 4<p<4<p<\infty unless the latter condition is satisfied. It is long known that the high energy parts are bounded in Lp(R4)L^{p}(\mathbb{R}^{4}) for all 1p1\leq p\leq\infty and that the same holds for W±W_{\pm} if HH has no eigenfunctions nor resonances at the threshold.