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.¹⁾ We always assume to start with a real-analytic initial state $\Psi_0$. Real-analytic means that at every point $x_0$ one can write the function locally as a power series (Taylor expansion). Because there is a positive convergence radius at every $x_0$ in real space, the area of convergence also extends into the complex domain. Thus real-analytic automatically means there is an open domain $D$ in the complexified configuration space $\mathbb{C}^n$ that includes the full real configuration space $\mathbb{R}^n$, i.e., $\mathbb{R}^n \subset D \subset \mathbb{C}^n$, and on which the function is holomorphic. It is yet not guaranteed that this domain $D$ includes a strip of finite width around the real configuration space as assumed in some related results about analyticity of the wave function.

The first positive result is space-analyticity under free evolution of an analytic initial state. Then it should be studied, if analyticity is conserved also with non-analytic potentials like the singular Coulomb potential.