Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities
Documenta mathematica, Tome 21 (2016), pp. 391-443
We consider the continuity property in Lebesgue spaces Lp(Rm) of the wave operators W± of scattering theory for Schrödinger operators H=−Δ+V on Rm, ∣V(x)∣≤C〈x〉−δ for some δ>2 when H is of exceptional type, i.e. N={u∈〈x〉sL2(Rm):(1+(−Δ)−1V)u=0}={0} for some 1/2−1/2. It has recently been proved by Goldberg and Green for m≥5 that W± are in general bounded in Lp(Rm) for 1≤p/2, for 1≤pall φ∈N satisfy ∫RmVφdx=0 and, for 1≤p∞ if ∫RmxiVφdx=0,i=1,...,m in addition. We make the results for p>m/2 more precise and prove in particular that these conditions are also necessary for the stated properties of W±. We also prove that, for m=3,W± are bounded in Lp(R3) for 13 and that the same holds for 1∞ if and only if all φ∈N satisfy ∫R3Vφdx=0 and ∫R3xiVφdx=0, i=1,2,3, simultaneously.
@article{10_4171_dm_537,
author = {K. Yajima},
title = {Remarks on $L^p$-boundedness of wave operators for {Schr\"odinger} operators with threshold singularities},
journal = {Documenta mathematica},
pages = {391--443},
year = {2016},
volume = {21},
doi = {10.4171/dm/537},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/537/}
}
K. Yajima. Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities. Documenta mathematica, Tome 21 (2016), pp. 391-443. doi: 10.4171/dm/537
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