Let $\mathcal{F}$ denotes the Fourier transform.

It is known that $\mathcal{F}(e^{-4\pi^2 i t |x|^2})(\xi)= e^{i |\xi|^2/4t}{(4\pi i t)^{-n/2}} \ (x, \xi\in \mathbb R^n).$

My question is: what is $\mathcal{F}(e^{-4\pi^2 i t |x|^\alpha})(\xi)$ for some $\alpha >0, \alpha \neq 2$?

Motivation: This kernel appears in the evolution of fractional Schrödinger equation.