Take Schrödinger's equation on $\mathbb{R}$, $i\partial_t\psi(x,t)=H\psi(x,t)$. Assume that $\psi(x,0)$ has compact support. Using known integral formulas for the propagators, it is fairly straightforward to verify that
- in the free case ($H = p^2$), $\psi(x,t)$ extends to an entire function of $x$ for all $t\neq 0$,
- in the harmonic oscillator case ($H=p^2 + x^2$), $\psi(x,t)$ extends to an entire function of $x$ for all $t\notin \pi\mathbb{Z}$.
In these cases, the cause seems to be that at the specified times, the integral kernel is a holomorphic function of two variables. Is there a way to see such behavior more generally, for example, what about the hamiltonians $p^2 + x^2 + \lambda x^4$ or $\sqrt{p^2 + m^2}$? I can't find any references for this, the closest question on this site is here, but it deals with analyticity in time. I'm also allowing for the possibility that the formulation of the problem I'm implying is not quite right: for example, we may wonder only about real analyticity in $x$, or perhaps restrict the initial condition to be also smooth... Anything along these lines will do.
If anyone is wondering, I stumbled upon this question because showing a result like this is one way of seeing that the Schrödinger equation (for given $H$) has non-causal behavior, much like the heat equation.
