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On L p boundedness of wave operators for 4‐dimensional Schrödinger operators with threshold singularities
Author(s) -
Jensen Arne,
Yajima Kenji
Publication year - 2008
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdm041
Subject(s) - mathematics , bounded function , eigenvalues and eigenvectors , operator (biology) , gravitational singularity , resonance (particle physics) , schrödinger's cat , mathematical physics , mathematical analysis , combinatorics , physics , quantum mechanics , chemistry , biochemistry , repressor , transcription factor , gene
Let H =−Δ+ V ( x ) be a Schrödinger operator on L 2 ( R 4 ), H 0 =−Δ. Assume that | V ( x )|+|∇ V ( x )|⩽ C 〈 x 〉 −δ for some δ>8. Let W ± = s − l i m t → ± ∞e i t He − i t H 0be the wave operators. It is known that W ± extend to bounded operators in L p ( R 4 ) for all 1⩽ p ⩽∞, 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 L p ( R 4 ) for all p such that 4/3< p <4.