The L^p-boundedness of wave operators for nonhomogeneous fourth-order Schrödinger operators in high dimensions

2025/04/08 by Zijun Wan, Wan, Zijun, Xiaohua Yao +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2504.05635

openalex publication_date 2025/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper investigates the L^p-boundedness of wave operators associated with the nonhomogeneous fourth-order Schödinger operator H = Δ² - Δ+ V(x) on ℝⁿ. Assuming the real-valued potential V exhibits sufficient decay and regularity, we prove that for all dimensions n ≥ 5 , the wave operators W_±(H, H₀) are bounded on L^p(ℝⁿ) for all 1 ≤ p ≤ ∞ , provided that zero is a regular threshold of H . As applications, we derive the sharp L^p-L^p' dispersive estimates for Schrödinger group e^-itH, as well as for the solutions operators cos(t √(H)) and (sin (t √(H)))/( √(H)) associated with the following beam equations with potentials: ∂_t² u + (Δ² -Δ+ V(x) ) u = 0, u(0, x) = f(x), ∂_t u(0, x) = g(x), (t, x) ∈ ℝ × ℝⁿ, n≥5, where p' denotes the Hölder conjugate of p, with 1 ≤ p ≤ 2. Moreover, we remark that the same results hold for the operator εΔ² - Δ+ V with a parameter ε>0, providing greater flexibility for the analysis of related equations.

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