For $\xi_0\in\mathbf{R}^n$ compute the solution of the Schrodinger equation with initial data $$ i\partial_tu-\Delta u=0 \text{ in } (0,t)\times\mathbf{R}^n\\ u(0,x)=e^{i\xi_0\cdot x} \text{ for all } x\in\mathbf{R}^n. $$ Give an interpretation of the solution.
The only way that I know of finding the solutions to the Schrodinger equation is by taking the Fourier transform with respect to $x$ of the equation, solving it for $\hat{u}$, then using the convolution of the transform of the initial data with the transform of the complex heat kernel (Schrodinger kernel?). So when I try to do that I get this far.
$$ \mathcal{F}\{u(t,x)\}(k)=e^{i|k|^2t}\mathcal{F}\{e^{i\xi_0\cdot x}\} $$
My issue now is that the initial data is not Schwartz, and so its Fourier transform does not exist. When I try to take the Fourier transform, I have to evaluate a circular function at $\pm\infty$, which doesn't make sense to me.
Any advice would be much appreciated!
Tom
HINTS:
The Fourier transform is defined for all tempered distributions. In this generalized sense you have $$ \mathcal{F}_{x\to k}\left(e^{i\xi_0\cdot x}\right)=(2\pi)^{d}\delta(k-\xi_0), $$ where $\delta$ is the Dirac delta.
As for the interpretation, try thinking at the result of applying a Galilean transformation to the initial datum $u_0(x)=1$.
Note 1: A Galilean transformation, aka "Galilean boost", is the map that sends the initial datum $u_0(x)$ to $$\mathcal{F}^{-1}_{k\to x}\left( \hat{u}_0(k+\xi_0)\right).$$
Note 2: I am using the following convention for the Fourier transform: $$ \mathcal{F}_{x\to k}(f(x))=\int_{\mathbb{R}^d}f(x)e^{-ix\cdot k}\, dx.$$