The Lp-continuity of wave operators for higher order Schrödinger operators

We consider the higher order Schrödinger operator H=(−Δ)m+V(x) in n dimensions with real-valued potential V when n>2m, m∈N, m>1. When n is odd, we prove that the wave operators extend to bounded operators on Lp(Rn) for all 1≤p≤∞ under n and m dependent conditions on the potential analogous to...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2022-08, Vol.404, Article 108450
Main Authors: Erdoğan, M. Burak, Green, William R.
Format: Article
Language:English
Subjects:
Higher order Schrödinger
Wave operator
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Summary:We consider the higher order Schrödinger operator H=(−Δ)m+V(x) in n dimensions with real-valued potential V when n>2m, m∈N, m>1. When n is odd, we prove that the wave operators extend to bounded operators on Lp(Rn) for all 1≤p≤∞ under n and m dependent conditions on the potential analogous to the case when m=1. Further, if V is small in certain norms, that depend n and m, the wave operators are bounded on the same range for even n. We further show that if the smallness assumption is removed in even dimensions the wave operators remain bounded in the range 1
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2022.108450