2024/01/10 by Cheng Han, Shanlin Huang, Cheng, Han +5
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2401.04969
openalex publication_date 2024/01/10 · openalex created_date 2024/01/13 · openalex updated_date 2026/07/28
In this paper, we study the fundamental solution of the higher order Schrödinger equation i∂t u(x,t) = ((-Δ)m + V(x))u(x,t), t ∈ ℝ, x ∈ ℝn, for any odd dimension n and integer m ≥ 1 satisfying n < 4m, where V is a real-valued bounded potential with suitable decay. Let Pac(H) denote the projection onto the absolutely continuous spectral subspace of H = (-Δ)m + V, and assume H has no positive embedded eigenvalues. Our main result says that the evolution operator e-itHPac(H) has an integral kernel K(t,x,y) satisfying the pointwise estimate |K(t,x,y)| ≤ C (1 + |t|)-h (1 + |t|-(n)/(2m)) (1 + |t|-(1)/(2m)|x - y|)-(n(m-1))/(2m-1), t ≠ 0, x,y ∈ ℝn, where the exponent h depends on m, n, and the zero energy resonance structure of H. We also prove analogous estimates for smoothing operators of the form H^\fracα2me-itHPac(H). The key innovation of this paper is a unified approach to deriving asymptotic expansions of the perturbed resolvents around zero, which comprehensively addresses all possible resonance types.