https://mage.uber.space/dokuwiki/ 2026-05-03T18:55:54+00:00 https://mage.uber.space/dokuwiki/ https://mage.uber.space/dokuwiki/lib/tpl/dokuwiki/images/favicon.ico text/html 2017-01-24T14:42:37+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/space-analyticity_of_wave_function/coulomb_potential?rev=1485265357&do=diff Space-analyticity of wave function under evolution with Coulomb potential Free evolution conserves analyticity but what if the initial state $\Psi_0$ is real-analytic but the acting Hamiltonian includes a non-analytic or even singular potential, like the Coulomb central potential $V(x) = -1/|x|$$t \neq 0$$\mathcal{C}^1$$V$$V$$V$$E(t, x, y)$$\mathrm{e}^{-\mathrm{itH_0}}$$E(t, x, y)$$t\neq 0$$V$$V$$|x|$ text/html 2016-11-25T15:19:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/space-analyticity_of_wave_function/free_evolution?rev=1480083594&do=diff 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 $\Psi(t,x)$\begin{equation}\label{eq-schroedinger} \mathrm{i}\partial_t \Psi = -\frac{1}{2}\Delta\Psi \end{equation}$\hbar = m = 1$$L^2$$f$$L^2(\mathbb{R}^n)$$\mathrm{e}^{b|x|}f \in L^2(\mathbb{R}^n)$$b<a$$\hat{f… text/html 2017-01-10T17:48:47+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/space-analyticity_of_wave_function/start?rev=1484066927&do=diff Space-analyticity of wave function The object of this topical section is to discuss conditions such that a wave function $\Psi(t,x)$ as a solution to the time-dependent Schrödinger equation with initial state $\Psi_0$ is jointly real-analytic in all $n=3N$ space coordinates.$\Psi_0$$x_0$$x_0$$D$$\mathbb{C}^n$$\mathbb{R}^n$$\mathbb{R}^n \subset D \subset \mathbb{C}^n$$D$