====== Space-analyticity of wave function under free evolution ======

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}\label{eq-schroedinger}
\mathrm{i}\partial_t \Psi = -\frac{1}{2}\Delta\Psi
\end{equation}
where [[wp>Atomic_units|Hartree atomic units]] $\hbar = m = 1$ are used. A positive result is given in [(:cite:cwf:hayashi1990)] 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 [[wp>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 [(:cite:cwf:reed-simon-1)] (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 relates directly to the Paley-Wiener theorem which is given here in an extended version that can be found in [(:cite:cwf:reed-simon-1)] (Th. IX.13).

**Theorem.** Let $f$ be in $L^2(\mathbb{R}^n)$. Then $\mathrm{e}^{b|x|}f \in L^2(\mathbb{R}^n)$ for all $b<a$ if and only if $\hat{f}$ has an analytic continuation to the open strip (or rather cylinder) $S(a) = \{ k \in \mathbb{C}^n \mid |\Im k|<a \}$ with the property that for each $\eta \in \mathbb{R}^n$ with $|\eta|<a$, $\hat{f}(\cdot + \mathrm{i} \eta) \in L^2(\mathbb{R}^n)$ and for any $b<a$
\[
\sup_{|\eta|\leq b}\|\hat{f}(\cdot + \mathrm{i} \eta)\|_2 < \infty.
\]

**Corollary.** If for some $a>0$ the initial state $\Psi_0$ fulfils all the properties of $\hat f$ in the theorem above then the same properties hold for $\Psi$ as the solution of the free Schrödinger equation \eqref{eq-schroedinger} at all times.

**Proof.** We clearly set $\hat f = \Psi_0$ in the Paley-Wiener theorem above, then the inverse Fourier transform of $\Psi_0$ and thus also the direct Fourier transform $\hat \Psi_0$ (which is identical to the inverse one modulo a parity switch $k \mapsto -k$) has $\mathrm{e}^{b|k|}\hat\Psi_0 \in L^2(\mathbb{R}^n)$ for all $b<a$. Now the free evolution for a time period $t$ in Fourier space is just a multiplication with the phase factor $\mathrm{e}^{-\mathrm{i}tk^2/2}$ which will not change this $L^2$ condition. This means when taking the Fourier transform to go back to ordinary space the Paley-Wiener theorem becomes applicable once more and $\Psi(t)$ has just the same properties as the inital state. $\Box$

If the initial state is assumed analytic but the evolution is not free, including a non-analytic or even singular potential (like the Coulomb central potential), one would suppose that the analytic property gets destroyed. This case will be discussed on a [[coulomb_potential|separate page]].