Inverse-Fourier transform of a function after non-linear frequency modulation

Question

Suppose $g\in L^1(\mathbb{R})$ such that $\hat{g}\in L^1(\mathbb{R})$ too. So $\tilde{g}(x) = \int_{-\infty}^{\infty}e^{i\pi \xi^2}\hat{g}(\xi)e^{2\pi i \xi x}\,d\xi$ is well-defined. Question is: Is there any relation between $g$ and $\tilde{g}$?

Answer 1

Of course, there are many relations between these two functions, but possibly the most interesting one is the fact that this function will represent a fixed-time evolution of the original function according to the solution of the Schrödinger's IVP

$$ \begin{cases} \partial _t u = i \Delta u \\ u_0 = g \end{cases}$$

(This can be seen by observing that for fixed time $t$ we've that $\hat{u} (\xi,t) = e^{-4i\pi^2 \xi^2 t} \hat{g} (\xi) $)

For example, it can be shown that we may write

$$ \tilde{g} (x) = (-i)^{1/2} e^{-4 \pi i x^2} * g (x) $$