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

In this paper, for any odd n and any integer m1 with n>4m, we study the fundamental solution of the higher order Schr\"odinger equation equation* iₜu (x, t) = ( (-) ᵐ+V (x) ) u (x, t), t R, \, \, x Rⁿ, equation* where V is a real-valued C^n+1{2-2m} potential with certain decay. Let P₀₂ (H) denote the projection onto the absolutely continuous spectrum space of H= (-) ᵐ+V, and assume that H has no positive embedded eigenvalue. Our main result says that e^-itHP₀₂ (H) has integral kernel K (t, x, y) satisfying equation* |K (t, x, y) | C (1+|t|) ^- (n{2m-) } (1+|t|^-n{2 m}) (1+|t|^-1{2 m}|x-y|) ^-n (m-1) {2 m-1}, t0, \, x, yⁿ, equation* where =2 if 0 is an eigenvalue of H, and =0 otherwise. A similar result for smoothing operators H^2me^-itHP₀₂ (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.