Let  $S$ be a  nonnegative self-adjoint operator on a  complex Hilbert space $X$. (For example, $X$ consists of functions on $\mathbb R^d$; it could be $L^2(\mathbb R^d), \dot{H}^2(\mathbb R^d)$, etc.)

We consider the abstract Cauchy problem  for the Schrödinger equation (SE):
$$
i\frac{\partial}{\partial t} u(x,t) + Su(x,t)=0, u(x,0)=u_0(x).
$$

Formally, we  may write the solution of $(SE)$,
$$u(x,t)= e^{itS}u_0(x).$$

>**Question:** If $u_0(x)\in X$ is [radial](https://en.wikipedia.org/wiki/Radial_function), then is it true that $e^{itS}u_0(x)$ is also  radial?

**Note:**   
(1) I know that if $S=- \Delta$, then $e^{itS}u_{0}$ is radial  whenever $u_0$ is radial.   
(2)  Specifically, I am interested in $S=-\Delta+ \frac{a}{|x|^2}$, see  [Section 1.1](https://arxiv.org/abs/1503.02716) for details (with $d\geq 3, a\geq (\frac{d-2}{2})^2$).