Pointwise estimates for the fundamental solutions of higher order Schr\"{o}dinger equations in odd dimensions II: high dimensional case

In this paper, for any odd n and any integer m≥1 with $n>4m$, we study the fundamental solution of the higher order Schr\"{o}dinger equation {equation*} i∂_tu(x,t)=((-Δ)^m+V(x))u(x,t), t∈ R,\,\,x∈ R^n, {equation*} where V is a real-valued Cn+1/2-2m potential with certain decay. Let Pac(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-itHPac(H) has integral kernel $K(t,x,y)$ satisfying {equation*} |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^n, {equation*} where σ=2 if $0$ is an eigenvalue of H, and σ=0 otherwise. A similar result for smoothing operators H^α/2me-itHPac(H) is also given. The regularity condition V∈ Cn+1/2-2m is optimal in the second order case, and it also seems optimal when $m>1$.