I am trying to solve the equation $$iu_t+u_{xx}= 0$$ with initial condition $$u(x,0) = f(x)$$ I solved the equation using Fourier transforms and got $$u(x,t) = \int F(k)e^{-ik^2t}e^{-ikx}dk$$ However I would like to write this in terms of the influence function, using the convolution theorem. $F(k)$ is just the F.T. of the $f(x)$, but I am looking for the function whose $F.T.$ is given by $e^{-ik^2t}$.
I know that if $$F(k) = e^{-ak^2}$$ then the inverse Fourier transform is $$f(x) = \sqrt{\frac{\pi}{a}}e^{x^2/(4a)}$$In my example can we just let $a = it$? Then the F.T. is $\sqrt{\frac{\pi}{it}}e^{x^2/(4it)}$. Is this correct?