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--- cwf:space-analyticity_of_wave_function:free_evolution [2016/11/22 13:49] – created markus
+++ cwf:space-analyticity_of_wave_function:free_evolution [2016/11/25 15:19] (current) – markus
@@ Line 1: / Line 1: @@
+====== 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]].