Free evolution means a solution to the time-dependent Schrödinger equation without any potential or interaction. For an initial state $\Psi_0(x)$ that is assumed to be space-analytic one has to check space-analyticity of solutions $\Psi(t,x)$ to \begin{equation} \mathrm{i}\partial_t \Psi = -\frac{1}{2}\Delta\Psi \end{equation} where Hartree atomic units $\hbar = m = 1$ are used. A positive result is given in ^[1] for one-dimensional configuration space with an additional exponential decay assumption on the initial state. The resulting wave function is not only space-analytic but even entire (holomorphic on the whole complex plane). A much simpler proof for space-analyticity alone can be constructed if one employs the Paley–Wiener theorem that gives a criterion for analyticity using decay properties with respect to the $L^2$ norm of the Fourier transform. That means the distribution of the frequency components controls if a function is analytic or not. Such a theorem is especially intriguing because as Reed and Simon phrase it in ^[2] (p. 382),

“It is not evident from looking at the basic definitions in functional analysis that the theory of analytic functions should play any role in the subject at all. That complex variable techniques are applicable is due mainly to the analyticity of the resolvent and to the analyticity properties of the Fourier transforms of functions with restricted support.”

The last statement realtes directly to the Paley-Wiener theorem.

Theorem (Paley-Wiener) Let $f$ be in $L^2(\R^n)$. Then $\e^{b|x|}f \in L^2(\R^n)$ for all $b<a$ if and only if $\hat{f}$ has an analytic continuation to the set $\{ k \mid |\Im k|<a \}$ with the property that for each $\eta \in \R^n$ with $|\eta|<a$, $\hat{f}(\cdot + \i \eta) \in L^2(\R^n)$ and for any $b<a$ \[ \sup_{|\eta|\leq b}\|\hat{f}(\cdot + \i \eta)\|_2 < \infty. \]

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