L^p-boundedness of wave operators for fourth order Schrödinger operators with zero resonances on ℝ³

2023/11/12 by Mizutani, Haruya, Wan, Zijun, Yao, Xiaohua
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2311.06763

Abstract

Let H = Δ² + V be the fourth-order Schrödinger operator on ℝ³ with a real-valued fast-decaying potential V. If zero is neither a resonance nor an eigenvalue of H, then it was recently shown that the wave operators W_±(H, Δ²) are bounded on L^p(ℝ³) for all 1 < p < ∞ and unbounded at the endpoints p=1 and p=∞. This paper is to further establish the L^p-boundedness of W_±(H, Δ²) that exhibit all types of singularities at the zero energy threshold. We first prove that W_±(H, Δ²) are bounded on L^p(ℝ³) for all 1 < p < ∞ in the first kind resonance case, and then proceed to establish for the second kind resonance case that they are bounded on L^p(ℝ³) for all 1 < p < 3, but not if 3 ≤ p ≤ ∞. In the third kind resonance case, we also show that W_±(H, Δ²) are bounded on L^p(ℝ³) for all 1

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