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Proceedings of the London Mathematical Society Advance Access originally published online on October 19, 2007
Proceedings of the London Mathematical Society 2008 96(1):136-162; doi:10.1112/plms/pdm041
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© 2007 London Mathematical Society

On Lp boundedness of wave operators for 4-dimensional Schrödinger operators with threshold singularities

Arne Jensen

Department of Mathematical Sciences
Aalborg University, Fredrik Bajers Vej 7G
DK-9220 Aalborg Ø
Denmark

Kenji Yajima

Department of Mathematics
Gakushuin University
1-5-1 Mejiro, Toshima-ku
Tokyo 171-8588
Japan
kenji.yajima{at}gakushuin.ac.jp

Received 14 January 2007.

Let H=–{Delta}+V(x) be a Schrödinger operator on L2(R4), H0=–{Delta}. Assume that |V(x)|+|{nabla} V(x)|≤C < x >–{delta} for some {delta}>8. Let Formula be the wave operators. It is known that W± extend to bounded operators in Lp(R4) for all 1≤p≤{infty}, if 0 is neither an eigenvalue nor a resonance of H. We show that if 0 is an eigenvalue, but not a resonance of H, then the W± are still bounded in Lp(R4) for all p such that 4/3<p<4.

2000 Mathematics Subject Classification 35J10 (primary), 47A40 (secondary).


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