Remarks on LpL^p-boundedness of wave operators for Schrödinger operators with threshold singularities

  • K. Yajima

Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities cover
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Abstract

We consider the continuity property in Lebesgue spaces Lp(Rm)L^{p}(\R^{m}) of the wave operators W±W_\pm of scattering theory for Schrödinger operators H=\lap+VH=-\lap + V on R\R^m, V(x)C\ax|V(x)|\le C\ax^-delta for some δ>2\delta>2 when HH is of exceptional type, i.e. \Ng=u\axsL2(Rm) ⁣:(1+(\lap)1V)u=00\Ng={u \in \ax^{s} L^{2}(\R^{m}) \colon (1+ (-\lap)^{-1}V)u=0 }\not={0} for some 1/2<s<δ1/21/2<s<\delta-1/2. It has recently been proved by Goldberg and Green for m5m\ge 5 that W±W_\pm are in general bounded in Lp(Rm)L^{p}(\R^{m}) for 1p<m/21\le p<m/2, for 1p<m1\le p<m if all \f\Ng\f\in \Ng satisfy RmV\fdx=0\int_{\R^{m}} V\f dx=0 and, for 1p<1\le p<\infty if RmxiV\fdx=0,i=1,,m\int_{\R^{m}} x_{i} V\f dx=0, i=1, \dots, m in addition. We make the results for p>m/2p>m/2 more precise and prove in particular that these conditions are also necessary for the stated properties of W±W_\pm. We also prove that, for m=3,W±m=3, W_\pm are bounded in Lp(R3)L^{p}(\R^{3}) for 1<p<31<p<3 and that the same holds for 1<p<1<p<\infty if and only if all \f\Ng\f\in \Ng satisfy R3V\fdx=0\int_{\R^{3}}V\f dx=0 and R3xiV\fdx=0,i=1,2,3\int_{\R^{3}} x_{i} V\f dx=0, i=1, 2, 3, simultaneously.

Cite this article

K. Yajima, Remarks on LpL^p-boundedness of wave operators for Schrödinger operators with threshold singularities. Doc. Math. 21 (2016), pp. 391–443

DOI 10.4171/DM/537