2025/05/11 by Cheng, Han, Soffer, Avy, Wu, Zhao +1
#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.2505.07009
This paper investigates the Lp-bounds of wave operators for higher-order Schrödinger operators H = (-Δ)m + V on ℝn, with m ≥ 2 and real-valued decaying potentials V. Our main objective is to establish the sharp Lp-boundedness of the wave operators W_±(H; (-Δ)m) in the presence of all types of zero-resonance singularities, for all odd dimensions 1 ≤ n ≤ 4m - 1. Specifically, for odd n with 1 ≤ n ≤ 4m - 1, there exist mn types of zero resonances for H, along with a critical type kc (both depending on n and m). If zero is a regular point of H or a k-th kind resonance with 1 ≤ k ≤ kc, the wave operators W_±(H; (-Δ)m) are bounded on Lp(ℝn) for all 1 < p < ∞. If zero is a k-th kind resonance with kc < k ≤ mn, we show that the range of p-boundedness for W_±(H; (-Δ)m) narrows to 1 < p < pk, where pk = (n)/(n - 2m + k + kc - 1). Additionally, if zero is an eigenvalue of H (i.e., k = mn + 1), then W_±(H; (-Δ)m) are bounded on Lp(ℝn) for all 1 < p < (2n)/(n - 1). Furthermore, it is shown that the wave operators W_±(H; (-Δ)m) are unbounded on Lp(ℝn) for all pk < p ≤ ∞ if kc < k ≤ mn, and for all (2n)/(n - 1) < p ≤ ∞ if zero is an eigenvalue of H with a non-zero solution ϕ to Hϕ= 0 in \bigcaps < -(1)/(2) L2s(ℝn) ∖ L2(ℝn)(referred to as a p-wave resonance). The key idea of the proof is to reduce the Lp-unboundedness to establishing the optimality of time-decay estimates for eitHPac(H) in weighted L2 spaces.