= Solution
Use $\widehat f(\xi)=\int e^{-ix\cdot\xi}f(x)dx$ with inverse factor $(2\pi)^{-n}$. The <Fourier transform> of the <Schrödinger equation> gives $i\partial_t\widehat\Psi=|\xi|^2\widehat\Psi/2$, hence $\widehat\Psi(\xi,t)=e^{-it|\xi|^2/2}\widehat f(\xi)$. To invert the multiplier, first damp it by $e^{-\varepsilon|\xi|^2}$. The <Gaussian integral> in each coordinate gives
$$
K_{t,\varepsilon}(x)=\bigl(4\pi(\varepsilon+it/2)\bigr)^{-n/2}
\exp\left[-\frac{|x|^2}{4(\varepsilon+it/2)}\right].
$$
Let $\varepsilon\downarrow0$, with the branch continuous from positive real damping. The <fundamental solution of a linear differential operator> is
$$
\boxed{K_t(x)=(2\pi it)^{-n/2}e^{i|x|^2/(2t)},\qquad
\Psi(x,t)=\int_{\mathbb R^n}K_t(x-y)f(y)dy.}
$$
For Schwartz initial data this <convolution> and Fourier derivation are justified directly; the multiplier tends to one as $t\downarrow0$ and has modulus one, giving the initial data and a unitary extension to $L^2$. The kernel itself is interpreted through the displayed Abel regularization.
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