Differences

This shows you the differences between two versions of the page.

--- cwf:space-analyticity_of_wave_function:free_evolution [2016/11/22 17:30] – markus
+++ cwf:space-analyticity_of_wave_function:free_evolution [2016/11/25 15:19] (current) – markus
@@ Line 2: / Line 2: @@
 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}
+\begin{equation}\label{eq-schroedinger}
 \mathrm{i}\partial_t \Psi = -\frac{1}{2}\Delta\Psi
 \end{equation}
@@ Line 9: / Line 9: @@
 > "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.
+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 (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 $b<a$
+**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 $b<a$
 \[
-\sup_{|\eta|\leq b}\|\hat{f}(\cdot + \i \eta)\|_2 < \infty.
+\sup_{|\eta|\leq b}\|\hat{f}(\cdot + \mathrm{i} \eta)\|_2 < \infty.
 \]
-//add references//
+**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]].