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.


This is known as the Kohlrausch-Williams-Watts function. There is no exact closed-form expression.

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).

There exists an analytic approximation, described in Relationship between the time-domain Kohlrausch-Williams-Watts and frequency-domain Havriliak-Negami relaxation functions (1991).


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