# Fractional Schrödinger equation

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.

The numerical evaluation for $$n=1$$ has been studied in Fourier Transform of the Stretched Exponential Function: Analytic Error Bounds, Double Exponential Transform, and Open-Source Implementation libkww (2009).