Exercise 18, § 1

(a) Let $n\in \mathbf{N}$ and let $f\in \mathscr{S}(\mathbf{R}^n)$. The function from $\mathbf{R}\times \mathbf{R}^n$ into $\mathbf{C}$ defined by

$u(t, x) =\int_{\mathbf{R}^n}\widehat{f}(y)$ exp(2$\pi ix\cdot y-4\pi^2t\|y\|^2$)$dy$

for $(t, x)\in \mathbf{R}\times \mathbf{R}^n$ satisfies

(1)

(“Schrödinger equation”) where

(b) The function $u$ is the unique solution of (1) such that $t\mapsto (x\mapsto u(t, x))$ belongs to $\mathscr{C}^{\infty}(\mathbf{R},\mathscr{S}(\mathbf{R}^n))$.