On wave operators for Schrödinger operators with threshold singuralities in three dimensions

2016/06/11 by Yajima, Kenji
#35P25 #47A40 #81U05 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1606.03575

Abstract

We show that wave operators for three dimensional Schrödinger operators H=-Δ+ V with threshold singularities are bounded in L¹(\mathbb R³) if and only if zero energy resonances are absent from H and the existence of zero energy eigenfunctions does not destroy the L¹-boundedness of wave operators for H with the regular threshold behavior. We also show in this case that they are bounded in L^p(\mathbb R³) for all 1≤ p ≤ ∞ if all zero energy eigenfunctions ϕ(x) have vanishing first three moments: ∫_\mathbb R³ x^αV(x)ϕ(x)dx=0, |α|=0,1,2.

Related

• The L^p boundedness of wave operators for Schrödinger operators with threshold singularities II. Even dimensional case
• L^p-boundedness of wave operators for fourth order Schrödinger operators with zero resonances on ℝ³
• L^p boundedness of wave operators for higher order schrödinger operators with threshold eigenvalues
• The L^p-boundedness of wave operators for two dimensional Schrödinger operators with threshold singularities
• Remarks on L^p-boundedness of wave operators for Schrödinger operators with threshold singularities