Pointwise estimates for the fundamental solutions of higher order Schrödinger equations in odd dimensions II: high dimensional case

2024/08/28 by Cheng, Han, Huang, Shanlin, Huang, Tianxiao +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.00117

Abstract

In this paper, for any odd n and any integer m≥1 with n>4m, we study the fundamental solution of the higher order Schrödinger equation i∂_tu(x,t)=((-Δ)^m+V(x))u(x,t), t∈ ℝ, x∈ ℝⁿ, where V is a real-valued C^(n+1)/(2)-2m potential with certain decay. Let P_ac(H) denote the projection onto the absolutely continuous spectrum space of H=(-Δ)^m+V, and assume that H has no positive embedded eigenvalue. Our main result says that e^-itHP_ac(H) has integral kernel K(t,x,y) satisfying |K(t, x,y)|≤ C(1+|t|)^{-((n)/(2m)-σ)}(1+|t|^{-(n)/(2 m)})(1+|t|^{-(1)/(2 m)}|x-y|)^{-(n(m-1))/(2 m-1)}, t≠0, x,y∈ℝⁿ, where σ=2 if 0 is an eigenvalue of H, and σ=0 otherwise. A similar result for smoothing operators H^\fracα2me^-itHP_ac(H) is also given. The regularity condition V∈ C^(n+1)/(2)-2m is optimal in the second order case, and it also seems optimal when m>1.

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